Equivalent equational identities which are not alphabetical variants of each other.

Question

Let our signature be that of a single binary operation $+$. Suppose $E$ and $E'$ are equivalent equational identities. Suppose also that neither $E$ nor $E'$ are equivalent to the trivial identity $x=y$, the reflexive identity $x=x$, or the constant identity $x+y=z+w$. Must $E$ and $E'$ be alphabetical variants of each other? Or is there a counterexample where there exist identities $E$ and $E'$ which are equivalent, but not alphabetical variants of each other?