I've recently been doing a search for these (A polychoron is a 4D polytope made up of polyhedra, like a polyhedra 3D made up of polygona)
The properties are:
1) Strictly-convex from body to cell (this is generalising it to higher dimensions)
2) Non-vertex transitive
3) Regular faced
I found 193 of these so far and hope to find more
They're based on the Johnson Solids
Why did the chicken cross the Mobius Band?
To get to the other ...um...!!!