  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 order $\mathcal{O}$. 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 both (Castryck, Houben,
Merz, Mula, Buuren, Vercauteren, 2023) and (De Feo, Fouotsa, Panny, 2024).
