Thinking about the post immediately before this one led me to see if I could connect opposite triangular faces of a rhombicosidodecahedron to form a ten-part compound — and it worked with Stella 4d just as it had when I “previewed” it in my head.

The interesting dual of the above polyhedral compound, also a ten-part compound, I was not able to preview in my head (although that would be a nice ability to have), but creating it was easy with Stella:

It is difficult, in the dual, to tell what the ten components are. To help with this, in the next image, all but one component has been removed. This reveals the components of the dual to be rhombus-faced parallelopipeds which are quite flattened, compared to most parallelopipeds I have seen before. This polyhedron is isomorphic to the cube, just as the elongated octahedra in the first compound were each isomorphic to the Platonic octahedron. Given that the cube and octahedron are duals, this is no surprise.

Stella 4d may be tried for free, as a trial download, at this website: http://www.software3d.com/Stella.php.