The operation of permuting data among the vertices of a hypercube computer induces a set of paths from senders to receivers. Permutations with edge-disjoint paths are desirable for efficient communication. The authors give simple algebraic descriptions for large classes of permutations that induce edge-disjoint paths for the commercially popular 'e-cube' routing algorithm. The descriptions cover most useful edge-disjoint permutations, and are easily applied in practice. Many previous proofs in the literature that specific permutations are edge-disjoint fall out as simple corollaries of the present work. Some new applications of this framework are presented. The first application considered concerns Gray code embeddings: the others are motivated by the connection of the present results to switching networks.