The snub dodecahedron is chiral, meaning it appears in left- and right-handed forms. This faceted version, where the same set of vertices is connected in different ways (compared to the original), possesses the same property.
Chiral polyhedra can always be tranformed into interesting polyhedral compounds by combining them with their own mirror-images. If this is done with the polyhedron above, you get this result, presented with a different coloring-scheme.
Both of these images were created using Stella 4d: Polyhedron Navigator, and you may try it at www.software3d.com/Stella.php.
To make this, I started with the dual of the great rhombicosidodecahedron, a polyhedron known as the dysdyakis triacontahedron. I then augmented half of its faces with tall prisms (thereby creating the chirality in this polyhedron), and took the convex hull of the result. The sixty red triangles are the tops of the augmentation-prisms.
A stellation of the above polyhedron, and a color-change, produced this result, also chiral. It may be enlarged with a click.
These polyhedra were created using Stella 4d, a program which you may buy — or try for free, as a trial download — at http://www.software3d.com/Stella.php.