There is a tension in homological approaches to birational geometry between reflexive modules and Cohen-Macaulay modules. On one hand, I will explain how CM modules are "not enough". For example, I will explain how to describe (and visualise) the stability manifold for 3-fold flops using reflexive modules, and outline some of the many geometric applications. On the other hand, in the viewpoint very much pushed by Ragnar, the much smaller class of CM modules should still "see" everything. In this flops setting, the tension is real: if you believe the CM philosophy, some of the statements you end up with are quite bizarre. As always in this story, CM modules turn out to win, but perhaps in an unsatisfactory way. Some parts of this talk are joint with Iyama, some with Hirano, some with Donovan, and some with Karmazyn.