We introduce a new tool for the study of isogeny-based cryptography, namely pairings which are sesquilinear (conjugate linear) with respect to the \(\mathcal {O}\) -module structure of an elliptic curve with CM by an imaginary quadratic field \(\mathcal {O}\) . We use these pairings to study the security of problems based on the class group action on collections of oriented ordinary or supersingular elliptic curves. This extends work of [CHM+23] and [FFP24].