Petrial tetrahedron

The Petrial tetrahedron is a regular skew polyhedron. It is composed of 3 skew squares It is the Petrie dual of the regular tetrahedron.

Hemicube
The hemicube is a tiling of the real projective plane which is abstractly equivalent to the Petrial tetrahedron. It's double cover is a cube and it can be seen as a cube with antipodal points identified. In other words it a quotient of the cube.

The hemicube is also related to the triangular hosohedron, as if the opposite edges and vertices of each square are identified, the result is a triangular hosohedron.

Related polyhedra
The rectified petrial tetrahedron is the tetrahemihexahedron, a uniform polyhedron.