It's gonna be a long answer, so kindly read fully to the end.
So, let's begin with Your questions.:
- Is there a Logic to these comparison.???
Yes, it's totally logical and authentic.
According to Vedas and Upanishads.:
"umaa sha~Nkarayogo yaH sa yogo vishhNuruchyate |" (Rudra Hridayopanishad 11)
"The combination of Uma and Sankara is Vishnu".(child).
"kaarya.n vishhNuH kriyaa brahmaa kaaraNa.n tu maheshvaraH |
prayojanaartha.n rudreNa muurtirekaa tridhaa kR^itaa |" (Rudra Hridayopanishad 15)
"The effect is Vishnu. The action is Brahma. The cause is Maheshwara. For the benefit of the worlds. Rudra has taken these three forms".
"yaa umaa saa svaya.n vishhNur" (Rudra Hridayopanishad 5)
"Uma Herself is in the form of Vishnu"
Rudra is the embodiment of all Devas. All devas are merely different manifestations of Sri Rudra Himself. On the right side of Rudra, there is the sun, then the four-headed Brahma, and then three Agnis (fires). On the left side, there exist Sri Umadevi, and also Vishnu and Soma (moon) (Rudra hridhaya upanishad )
Uma Herself is the form of Vishnu. Vishnu Himself is the form of the moon. Therefore, those who worship Lord Vishnu, worship Siva Himself. And those who worship Siva, worship Lord Vishnu. Those who envy and hate Sri Rudra, are actually hating Sri Vishnu. Those who decry Lord Siva, decry Vishnu Himself. (Rudra hridhaya upanishad )
Rudra is the generator of the seed. Vishnu is the embryo of the seed. Siva Himself is Brahma and Brahma Himself is Agni. Rudra is full of Brahma and Vishnu. The whole world is full of Agni and Soma. The masculine gender is Lord Siva. The feminine gender is Sri Bhavani Devi. All the mobile and immobile creation of this universe, is filled up with Uma and Rudra. The Vyakta is Sri Uma, and the Avyakta is Lord Siva. ( Rudra hridaya upanishad )
Same thing is represented in another way by Adi Shankara said in Shivananda Lahari as,
"Ardha vapusha baryatwam, Gonithwam sakkhitha..." (Shivananda Lahari Para-82)
"Lord Vishnu occupied half your body, Became transformed to be your wife (PARVATI), Became a boar to search for you, Became your lady friend to serve the nectar (MOHINI)"
Even puranas states the same.:
In the Kurma Purana, when Himavan praises Devi he says,
sahastrabāhuṃ puruṣaṃ purāṇam |
śayānamantaḥ salile tathaiva |
nārāyaṇākhyaṃ praṇato ‘smi rūpam ||” (kUrma PuraNa 1:11:245)
“I salute thy form called Narayana, O Lalita, which has a thousand heads, which is of infinite energy, having a thousand arms, the ancient Person, reclining on the waters”.
“goptri govindarupini |” (Lalita Sahasranama Stotram 1:63)
“[Lalitambika] assumes the form of Govinda for the task of protection”.
“vishnurupini |” (Lalita Sahasranama Stotram 1:166)
“Who is herself in the form of Vishnu”.
“mamiva pauruShaM rUpaM gopikAjanamohanam | kadAchidAdyA lalitA puMrUpA kR^iShNavigrahA | veNunAdavinodena vivashIkurute jagat |” (Brahmanda Puran)
"[Devi says] My male form as Krishna is bewildering the milk-maids. Krishna the manifestation of Lalita is beguiling entire jagat by his flute”.
Narayana(Vishnu) says to Virabhadra in Brahmanda Purana in the same section.
“ekaiva shaktiH parameshvarasya | bhinnA chaturdhA viniyogakAle |
bhoge bhavAnI samareShu durgA | kopeShu kALI puruSheshu viShNuH |”
“The ancient Sakti of the Lord (Parameshwara Shiva) appears in four forms; that Sakti becomes Bhavani (Mata Parvati) in her bhoga (grihasth) form, in battle she takes the form of Durga; in anger that of Kali; and she is present in her male form as me (Vishnu/Narayana).”
Lalita Saharanama Strotam.:
Karanguli nakhotpanna narayana dashakrutih | Mahapashu patastragni nirdagdhasura sainika |” (Lalita Sahasranama Stotram 1:32)
“Lalita – who out of her finger nails re-created all the ten Incarnations of Narayana(Vishnu) to destroy the Asuras slain by Him in His incarnations and now recreated magically by demon-Bhanda out of a missile of his.
She again burned to death the armies of demons with the fire of the great missile maha-pashupata”.
So, from above verses we can see that Lord Vishnu is the male form of Maa Uma.
- Where does Lord Shiva come into the picture.???*
The names ‘Hari’ and ‘Hara’ both have their origin from the root word ‘hru’, which means ‘to withdraw or deliver’. Therefore both the names Hari and Hara meanwithdrawal/deliverance. What do they withdraw? Hara withdraws all your sins and also the entire creation at the end of the time. That’s why Rudra is called as Hara. Vishnu delivers you of all your sins therefore he is Hari. This is only outer menaing.
In Hinduism, generally all female deities derive their names from their male counterparts. If we go by that rule, Bhava-Bhavani, Shiva-Shivani, Rudra-Rudrani, Kapardi-Kapardini, Maheshwara-Maheshwari, Parameshwara-Parameshwari… and likewise when Rudra is called as Hara, it is equally true that Bhavani can be called with the feminine sound of the word ‘hara’ as Hari. In fact it is Bhavani who delivers every one of sins as attested by the following verse from shruti.
"prapadye sharaNa.n devii.n du.ndurge durita.n hara |
taa.n durgaa.n durgamaa.n devii.n duraachaaravighaatiniim.h |
namaami bhavabhiito.aha.n sa.nsaaraarNavataariNiim.h |” (Devi Upanishad 1:28)
“Beyond Her is nothing; renowned is she as Durga; feared of life,I bow to Durga,Bulwark against all sins; the Pilot whoSteers me across the sea of worldly life”.
Therefore since Durga (Uma) delivers you through the sea of samsara by burning all your sins, she being the consort of Hara she is qualified to be called as HARI. So, ‘Hari’ is actually the name of Uma. Hope you understood now why Bhagawan Rudra has a name “hari-vallabha”, which means, “consort of Hari” which is same as "Umapati" or "Ambika Pati" .
So, the forms Ardhanareshwara = ShakaraNarayana (aka HariHara).
Sri Ishwara Gita Chapter 5 Vishwaroopam Darahanamam.
महादेवं महायोगं देवानामपि दैवतम् ।
पशूनां पतिमीशानं ज्योतिषां ज्योतिरव्ययम् ॥ ५.१२॥
पिनाकिनं विशालाक्षं भेषजं भवरोगिणाम् ।
कालात्मानं कालकालं देवदेवं महेश्वरम् ॥ ५.१३॥
उमापतिं विरूपाक्षं योगानन्दमयं परम् ।
ज्ञानवैराग्यनिलयं ज्ञानयोगं सनातनम् ॥ ५.१४॥
शाश्वतैश्वर्यविभवं धर्माधारं दुरासदम् ।
महेन्द्रोपेन्द्रनमितं महर्षिगणवन्दितम् ॥ ५.१५॥
आधारं सर्वशक्तीनां महायोगेश्वरेश्वरम् ।
योगिनां परमं ब्रह्म योगिनां योगवन्दितम् ॥
योगिनां हृदि तिष्ठन्तं योगमायासमावृतम् ।
क्षणेन जगतो योनिं नारायणमनामयम् ॥ ५.१६॥
ईश्वरेणैकतापन्नमपश्यन् ब्रह्मवादिनः ।
दृष्ट्वा तदैश्वरं रूपं रुद्रनारायणात्मकम् ।
कृतार्थं मेनिरे सन्तः स्वात्मानं ब्रह्मवादिनः ॥ ५.१७॥
Meaning: - Sages have darshana of Great God, Great Lord of Yoga, God of gods, Ishana, Pahupati, light of all lights (Supreme Light), holding Pinaka, remedies of all diseased beings, soul of Kala (Time), Kala of all the Kalas, Maheshwara, Umapati, having un-even eyes/vast eyes, Refuge of Knowledge and Detachment, eternal glorious, base of Dharma, hard to attain, worshiped by great Indra and Vishnu, praised by great rshis, source and base of all the energies, Lord of great yogis, Supreme Brahman, worshiped by yogis, situated in the heart of yogis, surrounded by yoga-maya. Sages of higher fortune witness the Unblemished Narayana who is in the form of yoni (Source/origin) of worlds getting one with Rudra. Sages consider themselves fortunate to witness the glorius form of Rudra allied to Narayana.
That’s why Rudrahridaya Upanishad states “rudro lingah uma peethah” means, “Rudra is Linga, and Uma is the the Base(Yoni)”.
That's why we chant Hari Om before starting any vedic mantras. It means when you chant Hari’s (Uma) name it would lead you to Omkara (Brahman).
Now, we know we is Omkara. Now, let’s see what shruti says about Omkara.
"atha hainaM bhaaradvaajaH paprachchha yaaj~navalkya.n ki.n taaraka.n ki.n taarayatiiti |" (Ramatapini Upanishad-Poorvopanishat 6.2)
"Sage Bharadwaja enquired Yajnyavalkya - 'What is Taraka? What is the one which liberates?"
"akaaraH prathamaaksharo bhavati ukaarodvitiiyaaksharo bhavati. makaarastR^itiiyaaksharo bhavati. ardhamaatrashchaturthaaksharo bhavati binduH pa~nchamaaksharo bhavati naadaH shhashhThaaksharo bhavati . taarakatvaattaarako bhavati |” (Ramatapini Upanishad-Poorvopanishat 6.3)
“That which has 'A' as the first syllable, 'U' as the second syllable, 'M' the third syllable, half-note is the fourth syllable, dot is the fifth syllable, and sound is the sixth syllable; that is called the Taraka”.
This Pranavam (Omkara) which is Taraka mantram is none other than Rudra alone as attested by following verses.
"nama staraya cha" (Yajurveda Sri Rudram Anuvaka-8)
"Salutations to Him (Rudra) who is the TARAKA (Pravana mantra - Om)".
Rudra is the Omkara because he alone delivers people from sins to liberation.
"namah paryaya chavaryaya cha | namah prataAranaya chottaranaya cha |" (Yajurveda Sri Rudram - Anuvaka:8)
"Salutations to Him who ferries men over the sins and evils of Samsara (the Illusions of the world), and who by the grant of knowledge ferries them over the Samsara altogether".
Further, the following verse states that there is no other way to liberation than knowing Rudra.
"sa eva sarvaM yadbhuutaM yachcha bhavyaM sanaatanam.h
j~naatvaa taM mR^ityumatyeti naanyaH panthaa vimuktaye "(Kaivalya Upanishad 9)
"He (Rudra) alone is all that was, and all that will be, the Eternal; knowing Him (Rudra), one transcends death; there is no other way to liberation".
So, finally we came to a point that it is Uma who is called by the name ‘HARI’ and it is Rudra who is ‘the ‘Omkara’. Therefore ‘Harih Om’ is another way to remember Bhavani-Shankara and is identical to say “Om Uma Maheshwarabhyam Namah”. And is also a way to remember the phrase “agnii somaatkmakam jagat”. Therefore Vedic seers remember ‘Uma and Maheshwara’ before beginning their Vedic works.
Epics narrated the story of Mohini incarnation of Vishnu only in the context of Milky Ocean churning. The son of Shiva and Mohini was named as ‘shAshThA’. Later on in south Indian folklore this ‘shAshThA’ got a more popular name as ‘Ayyappa’ (which means ‘Ayya’ + ‘Appa’ where both these words mean ‘father’ in Tamil) because he was a son born from two male gods (Shiva and Vishnu). He is also popular as ‘manikanTha’ in Tamil scriptures.
But in actual the Lord Ayyappa was the son of Shiva-Shakti only. Let me tell how. Both Mohini and Tripurasudnari are one and the same. Yes, you read it correctly; Mohini is none other than Tripurasundari. She was NOT an incarnation of Vishnu in reality.
‘Mohini’ is another synonym to mean ‘Tripurasundari’. This is why Shruti declaes goddess Tripurasundari as ‘Vishwa mohini’ (supreme enchantress of the universe). Refer to below evidence from Shruti.
"eshhaatmashaktiH | eshhaa vishvamohinii paashaaN^kushadhanurbaaNadharaa | eshhaa shriimahaavidyaa | " (Devi Upanishad 1:15)
"This is the power of Self, ‘vishwa-mohini’ (enchanting the universe), armed with the noose, the hook, the bow and the arrow. This is the great and holy Science".
So, how can this be possible.?
Its because when Uma who is Tripurasundari, transformed herself into a male form she is called Vishnu. What would happen if that male form undergoes a reverse transformation into the original female form? I am sure you would have guessed it correctly by now, that Vishnu’s transformation into female (mohini) is nothing but an esoteric way to stating that Vishnu displayed his original form as Tripurasundari. Note that Mohini was not a partial incarnation is was Vishnu himself fully transformed.
Brahmanda Purana says, when the pot of nectar emerged from ocean churning to possess it there started a fierce battle between deities and demons. Seeing that Vishnu transformed himself as a female enchantress as we read in Ramayana and Mahabharata and also the same is mentioned in all Puranas. However Brahmanda Purana clarifies it even further and says the following.
“EtasminanantarE vishNuh sarvalokaikarakshakah |samyagArAdhayAmAsa lalitAm svaikyarUpiNIm |” (Brahmanda Purana: Lalitopakhyana: 2:04)
“Meanwhile, Vishnu began to meditate on Sri Lalita Mahatripurasundari, who was his very own Self”.
“bhagavAnapi yogIndrah samAradhya mahEshvarIm |tadEkadhyAnayogEna tadrUpah samajAyat | sarvasammOhinI saa tu sAkshAt shrungAranAyika |sarvashrungAraveshAdhya sarvAbharaNabhUshitA | surANAmasurAnAM cha nirvAya raNamadbhutam |mandasmitEna daitEyAn mOhayantI jagAd cha |”(Brahmanda Purana: Lalitopakhyana: 2:8-10)
“By completely identifying himself with Maheshwari, Vishnu manifested himself in her form. She looked as if she were the very personification of the sentiment of erotica. Decorated with all kinds of ornaments, she appeared in the battlefield and brought the battle to a sudden halt."
Now let’s see her appearance aspects also which are identical to Lalitambika.
“sImantarEkhavinyastasindUrasHrENi bhAsurA |sphurachhandrakalOTamsA madAllOlavilOchanA |” (Brahmanda Purana: Lalitopakhyana: 2:67)
“The parting of her hair was decorated with a vermilion mark. Her crown wasdecorated with the half-moon. Her eyes were filled with intoxication”.
It's clearly says that she has crown of half moon like her husband Lord Shiva. Plus her hair was decorated with vermilion (Sindur) which is a sign of a married woman.
So, that form was identical and was the form of Maa Uma Maheshwari only.
Madhu Kaithabha Killing.:
Mahabharata [Book 9 (Shalya Parva)-chapter 49] states, “Vishnu himself, having in days of yore slain the Asuras, Madhu and Kaitabha”.
Whereas, Mahabharata [Book 6 (Bhishma Parva)-chapter 23] states, “I bow to thee that art fond of battle! O Uma, Sakambhari, O thou that art white in hue, O thou that art black in hue, O thou that hast slain the Asura Kaitabha”.
Mahishasura Mardini Stotram.:
“Madhu-Madhure Madhu-Kaittabha-Gan.jini Kaittabha-Bhan.jini Raasa-Rate
Jaya Jaya He Mahissaasura-Mardini Ramya-Kapardini Shaila-Sute |” (Mahishasuramardini Stotram).
“Who is as Sweet as Honey, Who Subdued the Pride of the Demons Madhu and Kaitabha and destroyed them, indulging in the Din and Uproar of the Great Battle. Victory to You, the Destroyer of the Demon Mahishasura, Who has Beautiful Locks of Hair and Who is the Daughter of the Mountain”.
Sri Shiva Purana 3:11:40.:
प्रकृतिस्त्वं पुमान्रुद्रस्त्वयि वीर्य्यं समाहितम् । त्वन्नाभिपङ्कजाज्जातः पञ्च वक्त्रः पितामहः ॥ ४० ॥
40. You (Vishnu) are Prakṛti and Rudra is Puruṣa. The seed is deposited in you. The five-faced Brahmā was born of your umbilical lotus.
Even some mahapuranas and upa puranas like Shiv Puran, Kalika Puran etc described how when Maa Uma took the incarnation of Lord Krishna, then Lord Shiva became Devi Radha. Whe she became Lord Rama, the he became Mata Sita and so on. Even Mahabharata praises Lord Vishnu and Maa Durga both as the slayer of Madhu and Kaitabha on seperate occasion and Lord Krishna while saying the Sahastranam of Lord Shiva says him as "Om Ye Namah" means the one who is Mother Lakshmi. But, I think the above verses are more than enough to prove my point here.
Now, there's difference of opinion as to this mapping. The MundamAla Tantram (which is another ShAkta Agama) gives a slightly different list as follows.:
Krishnastu kAlikA sAkshAd rAmamurtishcha tArini |
VarAho bhuvanA proktA nrisimho bhairavishwari ||
DhumAvati vAmanah syachinna bhrigukulodbhavah |
KamalA matsyarupah syAt kurmastu bagalAmukhi ||
MAtangi bauddha ityeshA shodashi kalkirupini |
Here the mapping is as follows:
For your question about validation of Todala tantra, it also have base in Shiva Purana. Shiva Purana in 3.16 gives Dash Avatar of Shiva and Shakti.
3.16 Dash Avatar (10 Incarnations of Shiva)
Describing about the ten incarnations and their corresponding power (Shakti), Sutji said- The first incarnation of lord Shiva was as Mahakal and his Shakti was called Mahakali. Lord Shiva took his second incarnation as Tar and his Shakti was called 'Tara'. The third incarnation of Lord Shiva was as Bhuvaneshwar and his Shakti was called 'Bhuvaneshwari'. Lord Shiva took his fourth incarnation as 'Shodash' who was also known as 'Srividdyesh' and his Shakti was called 'Shodashi' or 'Shri'. Lord Shiva took his fifth incarnation as Bhairav and his Shakti was called 'Bhairavi'. The sixth incarnation of Lord Shiva is famous as 'Chhinamastak' and his Shakti by the name of 'Chhinamasta.Lord Shiva took his seventh incarnation as 'Dhoomvan' and his Shakti was known as 'Dhoomvati. The
eighth incarnation was as Baglamukh and his Shakti as Baglamukhi. The nineth incarnation of lord Shiva became famous as Matang and his Shakti as 'Matangi'. Lord Shiva took his tenth incarnation as 'Kamal' and his Shakti as Kamala. If these ten incarnation of Shiva are worshipped along with his ten Mahavidyas then a man attains salvation.
Todala Tantra connects these Dash avatar of Shiva Shakti to Dash Avatar of Lord Vishnu.
Thus, Lord Shiva is the consort here in this picture (Question) in his respective forms.
I hope this clears your queries.