I used software called Stella 4d: Polyhedron Navigator to make this shape, and you can try Stella for free, right here. Sometimes, I use a lot of polyhedral modifications with this program, simply looking for a solid which looks interesting. With some polyhedra found through this random-walk process, I can’t even remember exactly how I created them in the first place — and this is one of those times.
I do know that this polyhedron has cuboctahedral symmetry, also known as octahedral symmetry. If anyone can figure out more about it, especially how to construct it, you are invited to share your insights in a comment.