Is there an identity strictly between commutativity and the constant identity?

Question

Let our language be a single binary operation symbol $\{*\}$. The commutative identity is $(x*y)=(y*x)$. The constant identity is $(x*y)=(z*w)$. I wonder, is there a single identity that is strictly between these two identities? That is, is there an identity $E$ such that the constant identity implies $E$ but not the other way around, and $E$ implies the commutative identity but not the other way around?