Not quite regular polyhedra
Take a naive interpretation of regular polyhedra:
All vertices (including epsilon ball) congruent
All edges congruent
All faces congruent
We can now find interesting families by removing one requirement. For example the uniform polyhedra have all vertices and edges congruent, but not all faces, and their duals have faces and edges congruent, but not vertices.
Are there examples, or interesting families, of polyhedra where every pair of faces is congruent and every pair of vertices, but not every pair of edges?