I will discuss some recent work establishing the orderability of contact manifolds which arise as a quotient of an aspherically fillable manifold by a finite group action which extends (non-freely) to the filling. This generalizes the well known case of lens spaces. The main tool is a geometrically equivariant version of contact Floer cohomology parametrized by a higher categorical refinement of the Eliashberg—Polterovich relation on the contact isotopy group, which is locally constant away from the discriminant. This is joint work with Dylan Cant and Jun Zhang.