A first compound of 20 octahedra can be constructed from the vertices of icositruncated dodecadodecahedron uniform polyhedron and a second from the vertices of the great dirhombicosidodecahedron or great snub dodecicosidodecahedron (E. Weisstein, Aug. 29-30, 2023). They are duals of corresponding cube 20-compounds, the second of which is noted and depicted by Wenninger (1982, pp. 139-140).
The skeleton of the first compound is 
(where 
is isomorphic to the octaherdal
graph) and of the second compound is the great
dirhombicosidodecahedral graph.
Both are implemented in the Wolfram Language as PolyhedronData[
"OctahedronTwentyCompound",
n
]
for 
and 2.
These octahedron 20-compounds are illustrated above together with their cube 20-compound duals and common midspheres.
The common solids and convex hulls of these compounds are illustrated above.
