I found another brief reference to an atonement method in the book Israelite Religion - this time a Greek ceremony, and with an odd connection with figs.
I have Googled the topic and found a couple of descriptions in books. First, from "Problems with Atonement"
And in another book (The Impact of Yom Kippur on Early Christianity)
I'm not sure if there is any connection with Jesus cursing a fig tree because it wasn't bearing any fruit.
And as to why figs were deemed particularly significant in Greek purification/atonement rituals: this site makes some observations, and uses a lot of big words, but it still seems rather unclear.
Whether it has anything to do with the fact that, if one eats a significant amount of figs at one sitting, one will feel nearly empty of all intestinal impurities within about 24 hours, remains another mystery...