I made this .gif, of two dodecahedra orbiting a common center of mass, using a program called Stella 4d: Polyhedron Navigator. This program may be tried for free at http://www.software3d.com/Stella.php.
This image of three rhombicosidodecahedra “orbiting” a common center was made with Stella 4d, a program you may try for free at this website.
I’ve been trying to figure out for over a year how to make images like the one above, without having holes in the two polyhedra, facing each other. At last, that puzzle of polyhedral manipulation using Stella 4d (software available at this website) has been solved: use augmentation followed by faceting, rather than augmentation followed by simply hiding faces.