Two Polyhedra, One Vertex Set

Here is an example of two polyhedra which use the same set of 15 vertices but are quite different in appearance. The two polyhedra "nest" together to form a distorted hexagonal prism. It's apparent that the two parts share a set of vertices.

The 15 vertex set comes from a dissection of the dodecahedron. If 4 vertices are added above the 15 vertex set and four are added below the vertex set two different polyhedra are created with regular polygons for faces. Each polyhedron has 19 vertices,