There are a number of attractive cube 20-compounds that can be constructed by taking the duals of the octahedra in the two octahedron 20-compounds. The second of these was noted and depicted by Wenninger (1983, pp. 139-140).
Both are implemented in the Wolfram Language as PolyhedronData["CubeTwentyCompound",
n] for 
and 2.
The vertices of the first cube 20-compound lead to attractive an attractive dodecahedron 6-compound, cube 6-compound, and tetrahedron 50-compound, while the second lead to a cube 25-compound and tetrahedron 50-compound (E. Weisstein, Aug. 30, 2023).
The cube 20-compound is illustrated above together with its octahedron 20-compound dual and common midsphere.
The first and second compounds have common solids that have the connectivity of a disdyakis triacontahedron and deltoidal hexecontahedron, respectively, while their convex hulls are unnamed polyhedron.
