A number of four-tetrahedron compounds can be constructed by rotating about the center-centroid lines of each face. Three such compounds are shown above for rotations by angles
for ,
3, and 4.

The compounds illustrated above are implemented in the Wolfram Language as PolyhedronData["TetrahedronFourCompound",
n]
for ,
..., 6.

These tetrahedron 4-compounds are illustrated above together with their duals
and common midspheres.

The common solids and convex hulls are illustrated
above.

The net for the hull of the third compound is illustrated above, where for tetrahedra with unit edge lengths,