  We present a notion of mutation of hyperbolic polyhedra, analogous to
mutation in knot theory, and then present a general question about
commensurability of mutant pairs of polyhedra. We motivate that question with
several concrete examples of mutant pairs for which commensurability is
unknown. The polyhedra we consider are compact, so techniques involving cusps
that are typically used to distinguishing mutant pairs of knots are not
applicable. Indeed, new techniques may need to be developed to study
commensurability of mutant pairs of polyhedra.
