The swastika symbol is used in many ceremonies, such as the yagya, among others. What is its significance?
Also, what is the correct direction for the swastika?
This or its mirror image?
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Swastika (स्वस्तिक) = Su (सु) + Asti (अस्ति) + Ka(क)
So, Swasti means Be Auspicious or Be Welfare
and Swastika means Auspicious/Welfare Doer
Swastika is believed to be a symbol of good, auspicious and welfare from ancient times.
Swastika also means for eternal.
There are two types of Swastika:
Swastika is made of two lines, both lines are drawn and divided at right angles (900) and the heads/terminals of them are stretched in left to right direction in such a way that they don't touch/meet.
Note: only right Swastika is almost used and reverse (left) Swastika is believed inauspicious sometimes.
In Hinduism Swastika are drawn or made before starting of any work with believing that work will be good, auspicious and welfare and complete with success.
Four lines are symbol of as follows:
In ancient times, at the beginning of any good work, there is an tradition of writing Mangalacharana (Invocation) but it is not easy for everyone to write Mangalacharana. So, Rishis provided the concept/hypothesis of Swastika symbol so-that every works be prosperous with bliss.
Swastika is believed to be a symbol of (according to different scripture and scholars):
The bijakshara and swarup of mantra of Ganpati appears like Swastika symbol.
Besides, Swastika has also very significance in Jyotisha and Vastu Shashtra.
ॐ can be taken in the form of 卐. According to some scholars, in the beginning of knowledge of Lipi/script, it is created in liner form instead of circular form. In the transformation/conversion of ॐ in to script 卐 is naturally formed.
Similar to Om which is believed to be full with divine guna, auspicious and elimination of disturbance (Vighnaharta) as having the power of Utpatti(creation), Sthiti(state) and Laya(Dissolution), Swastika is also believed having the habitat of Paramatma with power of creation, state and dissolution but with secondary/lower art comparatively to AUM.
Some scholars believes Swastika as distorted form of AUM.
According to Acharya Abhinava Gupta, Swastika is indicative of Naad Brahma (नाद ब्रह्म) or Akshar Brahma (अक्षर ब्रह्म).
Some Dharmic Beliefs of Swastika:
Source/Reference: Bharat Discovery articles of Swastika
ॐ स्वस्ति न इन्द्रो वृद्धश्रवाः।
स्वस्ति नः पूषा विश्ववेदाः।
स्वस्ति नस्तार्क्ष्यो अरिष्टनेमिः।
स्वस्ति नो ब्रिहस्पतिर्दधातु ॥
ॐ शान्तिः शान्तिः शान्तिः ॥
The swastika is well-recognized as an important Hindu symbol. It represents God (the Brahman) in his universal manifestation, and energy (Shakti). It represents the four directions of the world (the four faces of Brahma). It also represents the Purushartha: Dharma (natural order), Artha (wealth), Kama (desire), and Moksha (liberation).
(Source: Swastika on Wikipedia).
It's also associated with the sun.
Even some theories suggest it represents the Big Bang. Even the four arms represent the four stages of life (Brahmacharya, Grahasth, Vanaprasth and Sannyas). And it is used as a medium for Moksha.
Sri Chinmoy talks about the meaning of Swastika here:
When seen in the vital plane, Agni has seven arms in the form of a swastika. ‘Swastika’ is an old Sanskrit word that is quite often associated with Agni. What does the swastika mean? The exoteric meaning is good luck, prosperity and success. The esoteric significance of the swastika is inner progress, inner achievement, inner fulfilment.
Also, looking at the etymological roots of the word Swastika gives a lot of beautiful insights.
The word Swastika (स्वास्तिक) is derived from the Sanskrit root स्वस्तिक meaning:
a kind of mystical mark on persons or things meaning good luck
a lucky object
the meeting of four roads
the crossing of the arms
a mansion or temple of a particular form with a terrace in front
a particular mode of sitting practised by Yogis
Hence, it can be referred from the meanings that a Swastika symbolizes good luck. The symbolism in context of the number four has already been answered by Ankit. In my opinion, it could also be an abstract symbol for meditation (the crossing over of palms, with one palm over the other, while sitting in meditation). It could also be an abstract representation for a temple.