Skew compound of four triangles (chiral cubic symmetry)

The  is a regular skew compound polygon made of four triangles. An origami sculpture of this compound is called the WXYZ.

Vertex coordinates
The has the same vertices as a cuboctahedron. With unit side length vertex coordinates are given by all permutations of
 * $$\left(\pm\frac{\sqrt6}{6},\,\pm\frac{\sqrt6}{6},\,0\right).$$

Trivia
This compound is related to another compound of four trigonal dihedra. The compound of dihedra is regular in the sense of being flag-transitive if viewed as an embedding in $$\mathbb{R}^3$$ with flat faces even though the order of symmetry is only half of the number of flags, because multiple abstract isomorphisms correspond to the same spatial symmetry (for example, the two faces of one of the trigonal dihedra being swapped corresponds to the identity element of the spatial symmetry). However, it is not flag-transitive if viewed as a spherical tiling because the stabilizer of each triangular face is chiral.