What is the reasons/significance of Shiva holding deer in his hand?
I cannot get sources for this, but I remember reading this somewhere and hearing it from a song. So this an opinionated answer. There may be some other esoteric meaning. This is the meaning I know.
The Saranga or deer represents mind. If you see the way a deer runs, it jumps and runs in zig zag motion all over the place to escape the predator. This jump and zig zag motion is compared with mind's chanchalatva or constant motion.
Lords Shiva holds it. Indicating that he himself has complete mastery over his mind and will also still the mind of his Bhakta and helps a spiritual Sadhaka. As Yoga itself is defined as
Yogaha Chitta Vrutti Nirodha - Yoga Sutras 1.2
Yoga is the subduing and stopping of waves of mind
So it symbolizes that Lord Shiva has complete control over his mind and helps the yogi sadhaka or the spiritual aspirant to attain Yoga Or Union with the divine.
P.S: Just a FYI mention. In this song called Needu Shiva Needade Iru Shiva which means, "Oh Shiva either give or not give, it's your wish", from movie of Gaana Yogi Pachakshra Gawai, the song writer says,
Saranga Manake Nooraru Bayake Mundittu Ramisodu Na Kanene..!?
Don't I see you placing hundreds of desires before my mind which is like a deer, to tempt it?
So this meaning is a popular meaning in the culture.
The anectode of Daruka Vana Rishis could explain, how Lord Shiva came to hold the deer in His hand
In the temple at Chidambaram, on the ceiling of the large maṇḍapa in front of the shrine of Śivakāmasundarī, there is represented the story of the genesis of the Bhikṣāṭana form of Śiva and of Mohinī of Viṣṇu. The plan of Śiva and Viṣṇu to proceed to Dārukāvana to test the Ṛṣis and Ṛṣipatnis, by assuming the strange guise of the naked beggar and the bewitching enchantress, succeeds. Bhikṣāṭana captures the hearts of the Ṛṣipatnis while Mohinī ravishes the hearts of the sages, as narrated at length in long panels, row after row, in bright and beautiful colours. Beyond this story is that of the discomfited Ṛṣis performing the ābhicārika homa to destroy Śiva and how, out of the fire, arose the deer, the snake, the tiger, Muyalaka(Apasmara dwarf represent ing spiritual ignorance) and so forth
The deer(that came out of homa subdued by lord) on one side is the mind, because the mind leaps and jumps from one thing to another as unsteadily as that animal.
This sthala purana can be found in the website of chidambaram temple.
However it seems that , this sthala purana is not a part of any other puraana ( to my knowledge), and is exclusive to Chidambaram temple only.
PS Related post on Chidambara Maahatmya
What is the Symbolism of a Mriga (Deer).?
Rig Veda 1.80.7.:
Cloud-borne, Indra, wielder of the thunderbolt, verily your prowess is undisputed, since you, with (superior) craft, have slain that deceptive deer (māyinam mṛgaṃ), manifesting your own sovereignty.
Commentary by Sāyaṇa.: Ṛgveda-bhāṣya Deceptive deer.:
Vṛtra assumed the form of a deer.
So, here Vrtra = Mriga. But what is Vritra.?
Satapatha Brahmana 1:6:4:13.:
“An offering in honour of the Vritra-slayer, then, is the full-moon sacrifice. Vritra, assuredly, is no other than the moon; and when during that night (of new moon) he is not seen either in the east or in the west, then he (Indra) completely destroys him by means of that (new-moon sacrifice), and leaves nothing remaining of him. And, verily, he who knows this, overcomes all evil and leaves nothing remaining of evil”.
Before coming to why Vrtra is Moon let us first understand what moon stands for.
Rig Veda 10.19.13. (Purusha Shuktam).:
"The Moon was born from his (Veda Purusha's) mind."
Thus, as the moon was born from the mind of the Veda Purusha, it represents the mind (Manas).
So, finally we get Mriga (Deer) = Manas (Mind).
But, why Vrtra is said to be Manas or Moon.?
It's because all evils of Avidya are born out of mind (manas). Bhagawad Gita clearly states what evils one could face if mind is uncontrolled.
Bhagawad Gita 2.67.:
"For the mind, which follows in the wake of the wandering senses, carries away his discrimination as the wind carries away a boat on the waters."
And further it states how critical and important it is to control one’s mind, and it also tells us that with practice and with dispassion one’s mind can be controlled.
Bhagawad Gita 6.35.:
"Undoubtedly, O mighty-armed, the mind is difficult to control and is restless; but, by practice, O Son of Kunti, and by dispassion it is restrained."
What happens when one mind is subdued through practice (yoga) and knowledge.?
Mandala Brahmana Upanishad - 5.:
"The Manas influenced by worldly objects is liable to bondage; and that (Manas) which is not so influenced by these is fit for salvation. Hence the entire world becomes an object of Chitta; whereas the same Chitta when it is supportless and well-ripe in the state of Unmani, becomes worthy of Laya (absorption in Brahman). This absorption you should learn from me who am the all-full. I (Sun/Knowledge/Buddhi) alone am the cause of the absorption of Manas. The Manas is within the Jyotis (spiritual light) which again is latent in the spiritual sound which pertains to the Anahata (heart) sound. That Manas which is the agent of creation, preservation and destruction of the three worlds - that same Manas becomes absorbed in that which is the highest seat of Vishnu”.
So, after the mind is subdued (Unmani), the yogi becomes immortal by becoming free from the knots of the heart.
Now, this Unmani state has another name which is nothing but Turiya, the 4th.
Mandala Brahmana Upanishad 3.1.:
The great sage Yajnavalkya then asked the Purusha in the sphere (of the sun): “O Lord, though the nature of Amanaska has been defined (by you), yet I forget it (or do not understand it clearly). Therefore pray explain it again to me.” Accordingly the Purusha said: “This Amanaska is a great secret. By knowing this, one becomes a person who had done his duty. One should look upon it as Paramatman, associated with Sambhavi-Mudra and should know also all those that can be known through a (thorough) cognition of them. Then seeing Para-Brahman in his own Atman as the Lord of all (Sarvesham), the immeasurable (Aprameya), the birthless (Aja), the auspicious (Shiva), the supreme Akasa (Parama Akasha), the supportless (Niralamba), the secondless (Advaita) the only goal of Brahma, Vishnu and Rudra and the cause of all and assuring himself that he who plays in the cave (of the heart) is such a one, he should raise himself above the dualities of existence and nonexistence; and knowing the experience of the Unmani of his Manas, he then attains the state of Para- Brahman which is motionless as a lamp in a windless place, having reached the ocean of Brahmic bliss by means of the river of Amanaska-Yoga through the destruction of all his senses. Then he resembles a dry tree. Having lost all (idea of) the universe through the disappearance of growth, sleep, disease, expiration and inspiration, his body being always steady, he comes to have a supreme quiescence, being devoid of the movements of his Manas and becomes absorbed in Paramatman. The destruction of mans takes place after the destruction of the collective senses, like the cow’s udder (that shrivels up) after the milk has been drawn. It is this that is Amanaska. By following this, one becomes always pure and becomes one that has done his duty, having been filled with the partless bliss by means of the path of Taraka-Yoga through the initiation into the sacred sentences ‘I am pa’, 'That Thou Art', ‘I am thou alone’, ‘Thou art I alone’, etc.
Conclusion.: The symbolism behind Saguna Bhagawan Shiva holding the deer is that he is always in Turiya state.
Mandukya Upanishad 7.:
The Fourth (Turiya) is thought of as that which is not conscious of the internal world, nor conscious of the external world, nor conscious of both the worlds, nor dense with consciousness, nor simple consciousness, nor unconsciousness, which is unseen, actionless, incomprehensible, un-inferable, unthinkable, indescribable, whose proof consists in the identity of the Self (in all states), in which all phenomena come to a cessation, and which is unchanging fourth (Turiya), peaceful (Shanta), auspicious Shiva, and non-dual. That is the Self; that is to be known.
I hope this clarifies all your queries. Prd..
Here is one reason based a verse from halegannada poetry Rajashekhara vilasam(-by shadakshari) Lord brahma wanted to marry his own daughter saraswati. She was in the form of a deer.Brahma also took the form deer and was chasing her.That time lord Shiva cut Brahma's head.while Shiva held the symbol of deer in his hand as per Brahma's request. Here is the link to refer.[archive.org/details/rajashekhara].I'm not authentic in puranas or literature,however the same question also had bothered me a lot