Associocommutativity

The algebras satisfying $(a\cdot b)\cdot c=(a\cdot c)\cdot b$ for all $a,b,c \in A$ have been studied in geometry. For a special class, see our paper here. Denoting the right multiplication by an element $x$ by $R(x)$, we can rewrite the identity as $$ [R(x),R(y)]=R(x)R(y)-R(y)R(x)=0. $$ So the right multiplications all commute. There are several $K$-algebras which are neither commutative nor associative and they arise naturally in many areas of mathematics and physics.

First, note that, if $G$ is an abelian group acting on the right on a set $X$ (say we denote the action by $\cdot$), then we can get another right action $*$ of $G$ on $X$ by $x*g=x\cdot g^{-1}$.

Since addition and multiplication can be seen as abelian groups acting on themselves, this allows us to also view subtraction and division in this way.

Now, note that any right action $*$ of a commutative semigroup $(G,\cdot)$ on a set $X$ will have the property that you want: $(x*g)*h=x*(g\cdot h)=x*(h\cdot g)=(x*h)*g$.

This explains all your examples.