The Chasam Sofer (derashot second day Sukkkot) cites from the Rema M'pano that Atzeret is called חג related to the word חוגה, the midpoint that everything turns on for the kedusha of all the holidays stems from Atzeret. Why? For the holiday has no additional mitzvot, it is just about a התוועדות with Hashem and Klal Yisrael. That is the root of all the מועדים - to be a ועד - a time of meeting with Hashem.
It is noted that the word for a circle, מחול, and the word for forgiving, מחילה are the same root. On Yom Kippur we obtain forgiveness for our sins. Rav Levi Yitzchak teaches that Sukkot is the time of teshuva from love when sins turn into merits. We are then able to take the sins and turn them into a dance of a מחול on Simchas Torah.
This also what the Gemarah says at the end of Taanis that the tzaddikim will be sitting in a מחול, the complete atonement and joy together.
From the sefer שישו ושמחו סימן מט:
