This polyhedron has the same vertices as a 9-gon prism. Its faces are 9 rectangles, 18 isosceles triangles, and 18 self-intersecting tetragons. Its vertices are 9-valent; the sequence of faces around a vertex is 1 rectangle, 2 tetragons, 3 triangles, 2 tetragons, 1 rectangle.

(For clarity, I've shown only a few of the faces in addition to the rectangles. The other faces can be gotten by symmetry.)

There is no analogous polyhedron based on 7-gons or 5-gons. (The closest thing is the blend of a {7} prism and a {7/6} antiprism. But the analogue of that would be the blend of a {9} prism and a {9/8} antiprism, which is different from the polyhedron we have here.)

This suggests to me that there is no nice classification of non-convex isogonal polyhedra.

For example, what complicated polyhedra might exist with 35-gon prism symmetry?