Inverses of Elements Related by Compatible Relation

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $\mathcal R$ be a relation compatible with $\circ$.

Let $x, y \in G$.

Then
 * $x \mathrel{\mathcal R} y \iff y^{-1} \mathrel{\mathcal R} x$.

Proof
Let $e$ be the group identity of $G$.