BALTIC-WAY 2006 Problem 5

An occasionally unreliable professor has devoted his last book to a certain binary operation $*$. When this operation is applied to any two integers, the result is again an integer. The operation is known to satisfy the following axioms:

(a) $x *(x * y)=y$ for all $x, y \in \mathbb{Z}$;

(b) $(x * y) * y=x$ for all $x, y \in \mathbb{Z}$.

The professor claims in his book that

(C1) the operation $*$ is commutative: $x * y=y * x$ for all $x, y \in \mathbb{Z}$.

(C2) the operation $*$ is associative: $(x * y) * z=x *(y * z)$ for all $x, y, z \in \mathbb{Z}$.

Which of these claims follow from the stated axioms?