Medial layer graphs of equivelar 4-polytopes

In any abstract 4-polytope PP, the faces of ranks 1 and 2 constitute, in a natural way, the vertices of a medial layer graph GG. We prove that when PP is finite, self-dual and regular (or chiral) of type {3,q,3}{3,q,3}, then the graph GG is finite, trivalent, connected and 3-transitive (or 2-transitive). Given such a graph, a reverse construction yields a poset with some structure (a polystroma); and from a few well-known symmetric graphs we actually construct new 4-polytopes. As a by-product, any such 2- or 3-transitive graph yields at least a regular map (i.e. 3-polytope) of type {3,q}{3,q}.