SelfIntersecting Truncated QuasiRegular Polyhedra 
A polyhedron is truncated quasiregular if it is vertextransitive with scalene triangular vertex figures. Vertex transitivity means that for any two vertices of the polyhedron, there exists a translation, rotation, and/or reflection that leaves the outward appearance of the polyhedron unchanged yet moves one vertex to the other. A vertex figure is the polygon produced by connecting the midpoints of the edges meeting at the vertex in the same order that the edges appear around the vertex. There are only two truncated quasiregular polyhedra that are not selfintersecting, namely the Truncated Cuboctahedron and the Truncated Icosidodecahedron. When selfintersection is allowed, there are five other truncated quasiregular polyhedra.
Albert Badoureau and Johann Pitsch each described all five in 1881 [1][2].






References:  [1]  Jean Paul Albert Badoureau, Mémoire sur les Figures Isocèles, Journal de l'École polytechnique 49 (1881), 47172. 
[2]  Johann Pitsch, Über Halbreguläre Sternpolyeder, Zeitschrift für das Realschulwesen 6 (1881), 924, 6465, 7289, 216.  
[3]  H. S. M. Coxeter, M. S. LonguetHiggins, and J. C. P. Miller, Uniform Polyhedra, Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences 246 (1954), 401450. 