The dual of a truncated quasi-regular polyhedron is face-transitive with faces shaped like scalene triangles. Face transitivity means that for any two faces of the polyhedron, there exists a translation, rotation, and/or reflection that leaves the outward appearance of the polyhedron unchanged yet moves one face to the other. There are only two truncated quasi-regular duals that are convex and not self-intersecting, namely the Disdyakis Dodecahedron and the Disdyakis Triacontahedron. When non-convexity and self-intersection are allowed, there are five other truncated quasi-regular duals. Only one, the Great Disdyakis Dodecahedron, is not self-intersecting.