“Atonement of sins” is done through manifestation of the Light of Hochma (wisdom). The confession is the drawing of Hochma. The more one confesses, the more the Hochma appears on him. It is said about that: “and in that time, … the iniquity of Jacob shall be sought for, and there shall be none.” This is because for all the sin, when it is forgiven, it is not forgiven until Hochma is extended upon it. This is why they were looking for iniquities, to draw upon him the Light of Wisdom.
“The embrace of the left” means the extension of the left line. On each of the ten penitential days, one discernment of the ten Sefirot of Mochin de Hochma, called “left line,” is extended. And on Yom Kippur (Day of Atonement) is the Zivug (coupling).
The embrace of the right is the drawing of Hochma below the Chazeh (chest), the place of the manifestation, where it is already sweetened in Hassadim (mercy). It is primarily considered extending of Hassadim. The building of the Nukva itself continues until the eighth day of Sukkot, and on the eighth day is the Zivug.