The major avoda of Yom Kippur is making a lottery over two goats - one to Hashem and the other one to throw over the cliff לעזאזל - to azazel, the powers of evil. Which is the primary goat? The word לעזאזל appears four times in the parsha [and makes no other appearance in Tanach]. Four times לעזאזל is bi-gematria שעיר. Since the word שעיר has four letters each one represents לעזאזל. So the primary goat is לעזאזל - not the one offered in the beis hamikdash.
The gemara in maseches yoma says that the שעיר לעזאזל atones for מעשה עוזא ועזאל. These two angels descended into this world and were involved in illicit desires and פגם הברית [blemishing the bris]. As it says in parshas Beraishis הנפילים היו בארץ בימים ההם וגם אחרי כן אשר יבאו בני האלהים אל בנות האדם - The fallen ones were in the land in those days and also later, when the sons of the powerful came to the daughters of man. The way to fix an illicit desire is with fast - צום, as is alluded to in the words הנפילים היו בארץ the last letters of the three words are .... צום.
The פגם הברית corrupts the soul completely and that is why מעשה עוזא ועזאל is bi-gimatria 613. Yom Kippur atones for these תריג מצוות. On mincha of Yom Kippur we read the parsha of forbidden relations and when the kohen gadol went in to the holy of holies he wore the bigdei lavan which atone for פגם הברית [the bigdei zahav atone for avoda zara]. Specifically, the pants of the kohen gadol atone for revealed gilui arayos and the שעיר לעזאזל atones for hidden gilui arayos [meaning the penetration of the פגם הברית to the inner reaches of the soul, the deep subconscious].
The ultimate tikkun is not just the purity of Yom Kippur but reaching Kedushas Habris in married life, which is why Yom Kippur was one of the two days a year when the men would choose wives from among the dancing girls. There cannot be a Yom Kippur if the kohen gadol is not married - וכיפר בעדו ובעד ביתו - זו אשתו. If his wife dies on Yom Kippur he stops everything and marries another woman [Yerushalmi quoted by Tosfos]. The five immersions - ה' טבילות and the ten times he washes his hands and feet - י' קידושי הידים ורגלים symbolize the י and ה which are found in the words איש and אשה.
[Based on a shiur given by Rav Yitzchak Ginsburgh as recorded by R' Itiel Giladi]
This Torah should be a zchus for JAKEY BERMAN and DAHLIA HONIGSFELD upon their engagement. They are two of the most special and holiest people I know and it is apropos that they became engaged as we prepare for Yom Kippur. I owe each of them a tremendous debt of gratitude so this is my opportunity to say - Thank You.
MAZEL TOV!!!!:-)