The dual of the snub dodecahedron (above) is called the pentagonal hexacontahedron (below, left). The compound of the two is shown below, at right. (Any of the smaller images here may be enlarged with a click.)
Like all chiral polyhedra, both these polyhedra can form compounds with their own mirror-images, as seen below.
Finally, all four polyhedra — two snub dodecahedra, and two pentagonal hexacontahedra — can be combined into a single compound.
This polyhedral manipulation and .gif-making was performed using Stella 4d, a program you can find here.
As the dual of the snub dodecahedron, which is chiral, this member of the Catalan Solids is also chiral — in other words, it exists in left- and right-handed versions, known an entantiomers. They are mirror-images of each other, like left and right gloves or shoes. Here’s the other one, by comparison:
It is always possible to make a compound, for a chiral polyhedron, from its two enantiomers. Here’s the one made from the two mirror-image pentagonal hexacontahedra shown above:
Stellating this enantiomorphic-pair-compound twenty-one times produces this interesting result:
And, returning to the unstellated enantiamorphic-pair-compound, here is its convex hull:
This convex hull strikes me as an interesting polyhedron in its own right, so I tried stellating it several times, just to see what would happen. Here’s one result, after seventeen stellations:
Software credit: I made these rotating images using Stella 4d: Polyhedron Navigator. That program may be bought at http://www.software3d.com/Stella.php, and has a free “try it before you buy it” trial download available at that site, as well. I also used Geometer’s Sketchpad and MS-Paint to produce the flat purple-and-black image found on faces near the top of this post (and, by itself, in the previous post on this blog), but I know of nowhere to get free trial downloads of these latter two programs.