There is only one uniform polyhedral compound, the stellated octahedron, which has octahedral symmetry, order 48. It has a regular octahedron core, and shares the same 8 vertices with the cube.

If the edge crossings were treated as their own vertices, the compound would have identical surface topology to the rhombic dodecahedron; were face crossings also considered edges of their own the shape would effectively become a nonconvex triakis octahedron.

There are lower symmetry variations on this compound, based on lower symmetry forms of the tetrahedron.

• A facetting of a rectangular cuboid, creating compounds of two tetragonal or two rhombic disphenoids, with a bipyramid or rhombic fusil cores. This is first in a set of uniform compound of two antiprisms.
• A facetting of a trigonal trapezohedron creates a compound of two right triangular pyramids with a triangular antiprism core. This is first in a set of compounds of two pyramids positioned as point reflections of each other.

If two regular tetrahedra are given the same orientation on the 3-fold axis, a different compound is made, with D_3h, [3,2] symmetry, order 12.

Other orientations can be chosen as 2 tetrahedra within the compound of five tetrahedra and compound of ten tetrahedra the latter of which can be seen as a hexagrammic pyramid:

• Compound of cube and octahedron
• Compound of dodecahedron and icosahedron
• Compound of small stellated dodecahedron and great dodecahedron
• Compound of great stellated dodecahedron and great icosahedron

• Cundy, H. and Rollett, A. "Five Tetrahedra in a Dodecahedron". §3.10.8 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 139–141, 1989.

• Weisstein, Eric W. "Compound of two tetrahedra". MathWorld.
• Compounds of Polyhedra VRML model: [1]