Since I stumbled across this by stellating other polyhedra, I’ve never seen what a face of one component of this compound looks like, without having part of that face covered. My best guess is that the faces (of which there are eight in each part of the compound) are kites.

I used Stella 4d, available at http://www.software3d.com/Stella.php, to make this, and a free trial download is available at that site.