A symmetric set (also called an involutary quandle or a kei) is a set $A$ with a binary operation $\circ$ satisfying the following conditions for all $a,b,x\in A$:
- $a\circ a =a$;
- $(x\circ a)\circ a=x$;
- $x\circ(a\circ b)=(((x\circ b)\circ a )\circ b$.
See this paper for further details. Now my question is about the motivation of the last identity $$x\circ(a\circ b)=(((x\circ b)\circ a )\circ b.$$ What are some familiar binary operations that satisfy this identity?
One motivating example is the operation $x\circ a=ax^{-1}a$ on any group. In particular, when the group in question is $\mathbb{R}^n$, this operation $x\circ a=2a-x$ has a geometric interpretation: it is the "reflection of $x$ across $a$" (that is, $-\circ a$ is the negation map when you consider the point $a$ to be the origin). This geometric description can be generalized to make any symmetric space into a symmetric set, which I believe is the origin of the term "symmetric set".