Obscure properties of binary operations

Question

The commutative, associative, and distributive properties of binary operations are very well-known. I am interested in more obscure properties of binary operations, which have nonetheless been studied in the mathematical literature. By property of binary operations, I mean either a unary property of a single binary operation, or a property of an ordered pair of binary operations. For example, the "switchable" property which I came up with is this: Given two binary operations $+$ and $*$ on a set $S$, the ordered pair $(+,*)$ is said to satisfy switchability if, for all $x,y,z$ in $S$, $(x+y)*z=x+(y*z)$. I am interested in such obscure properties, especially those which have been studied in some math paper or text.