Name of property: $a \circ (a \circ b) = b$ How do you describe an operation like this?
$$ a \circ (a \circ b) = b $$
For example, XOR is like this:
$$ a \oplus a \oplus b = b $$
 A: In "A guide to self-distributive quasigroups, or latin quandles," David Stanovský calls such an operator left involutory.

A binary algebra $(A,*)$ is called left involutory (or left symmetric) if $x * (x * y) = y$ (hence we have unique left division with $x \backslash y = x ∗ y$).

It's not a very popular term—it currently only has 8 hits on Google—but it makes sense (since it means that the section function $(x * {})$ is involutory), and there are a couple of other papers which use the same terminology with the same meaning (and at least one of these papers cites Stanovský's paper).
So if you're going to write a paper using this term, I suggest using the term "left involutory," and consider citing Stanovský's paper when you do.
(Fun fact: one of the Google search results for "left involutory" caught my eye because it's a chat log from an IRC chat room that I speak in regularly! I couldn't resist mentioning this.)
A: If you can rearrange like this, $a \circ (a \circ b) = (a \circ a) \circ b$, then the expression you have implies that $a$ is both the left-inverse and the right-inverse of itself. Hence, $a$ is the inverse of itself.
