A large set of interconnection schemes, collectively referred to as multistage permutation networks, has emerged in the literature as a means for providing parallel communications in multiprocessing systems. It is known that most of these networks cannot realize every permutation over their terminal nodes. This fact has prompted research on functional relations among such networks. The correspondence deals with one aspect of this research, namely, the problem of constructing equivalence maps between two permutation networks. A branch-and-bound algorithm is given to test whether two such networks are equivalent in polynomial time. Whenever they are, the algorithm also determines a map that conjugates one network onto the other.