Quivers and their mutations play a fundamental role in the theory of cluster algebras. We focus on the problem of deciding whether two given quivers are mutation equivalent to each other. Our approach is based on introducing an additional structure of a cyclic ordering on the set of vertices of a quiver. This leads to new powerful invariants of quiver mutation. These invariants can be used to show that various quivers are not mutation acyclic, i.e., they are not mutation equivalent to an acyclic quiver. This talk is partially based on joint work with Sergey Fomin [arXiv:2406.03604].