Both of the polyhedra mentioned in the title are very interesting for a few reasons.
The Tetrahemihexahedron for one, is in the same regiment as the Octahedron, yet is not regular itself. No other uniform polyhedron exhibits this strange trait. Another thing is, it's dual is quite strange as well.
The Tetrahemihexacron is a compound, yet its dual is not. Another thing is that it has no caps at the end of its prisms, instead, there is a vertex at infinity. And most of all, it's regular, but it's dual isn't.