TITLE: Not quite regular polyhedra
QUESTION [7 upvotes]: 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?

REPLY [3 votes]: Allan Edmonds has a few papers studying "equifacetal simplices", i.e. simplices in which any two facets must be congruent.  Here are references:
Edmonds, Allan, The center conjecture for equifacetal simplices.  Adv. Geom. 9 (2009), no. 4, 563--576.
Edmonds, Allan L., The partition problem for equifacetal simplices.  Beitrage Algebra Geom. 50 (2009), no. 1, 195-213.
Edmonds, Allan, The geometry of an equifacetal simplex.  Mathematika 52 (2005), no. 1-2, 31-45.