- 1 | S4 x I / S4 x I (this is just the trivial case of 1 cube)
- 5 | A5 x I / A4 x I (the uniform compound of 5 cubes, 5-color version)(dual: uniform compound of 5 octahedra, 5-color version)
- 3 | S4 x I / D4 x I (the uniform compound of three cubes)(dual: 3 octahedra)(compound of compounds of cubes and octahedra)
- 4 | S4 x I / D3 x I (Bakos' compound) (dual: rigid uniform compound of 4 octahedra)
- n | D4n x I / D4 x I (n=5: D20 x I / D4 x I) (dual: 5 octahedra)
- n | D3n x I / D3 x I (n=2: D6 x I / D3 x I) (n=5: D15 x I / D3 x I) (dual: 5 octahedra)
- n | D2n x I / D2 x I (n=5: D10 x I / D2 x I) (dual: 5 octahedra)
- 6 | S4 x I / D2 x I (dual: 6 octahedra)
- 10 | A5 x I / D3 x I (version A) (version B) (20 cubes of versions A+B combined) (dual to A: uniform compound of 10 octahedra) (dual to B: uniform compound of 10 octahedra) (dual to A+B: 20 octahedra)
- 15 | A5 x I / D2 x I (dual: 15 octahedra)
- 6 | S4 x I / C4 x I (Skilling's uniform compound) (smaller separation angle) (dual: 6 octahedra)
- 2n | D4n x I / C4 x I (n=3: D12 x I / C4)
- 2n | D3n x I / C3 x I (n=1: D3 x I / C3 x I)
- 2n | D2n x I / D1 x I (n=1: D2 x I / D1 x I)
- nA | Dn x I / C2 x I (n=5: D5 x I / C2 x I)
- nB | Dn x I / C2 x I (n=5: D5 x I / C2 x I)
- 4 | A4 x I / C3 x I (dual: uniform nonrigid compound of 4 octahedra)
- 6 | A4 x I / C2 x I
- 8 | S4 x I / C3 x I (dual: uniform compound of 8 octahedra)
- 12 | S4 x I / D1 x I (varying angles: 1, 2, 3, 4, 5)
- 12A | S4 x I / C2 x I
- 12B | S4 x I / C2 x I
- 20 | A5 x I / C3 x I (varying angles: 1, 2, 3) (dual to 1: the rigid uniform compound of 20 octahedra meeting 2 per vertex), (duals to 2 and 3 are uniform compounds of 20 octahedra with rotational freedom)
- 30A | A5 x I / C2 x I (varying angles: 1, 2, 3, 4, 5)
- 30B | A5 x I / C2 x I (varying angles: 1, 2, 3, 4, 5)
- n | Cn x I / E x I (n=5: C5 x I / E x I)
- 2n | Dn x I / E x I (n=5: D5 x I / E x I)
- 12 | A4 x I / E x I
- 24 | S4 x I / E x I
- 60 | A5 x I / E x I
- Four cubes (S4 x I / D4 x I plus original)
- "Theosophical" Compound (5 cubes: S4 x I / D3 x I plus original)
- The cube of 14 cubes (all the rigid S4 cubes combined) (dual: 14 octahedra)

The notation can be understood approximately as follows: "*n* |
*P* x I / *Q* x I" indicates a compound of *n* cubes. The
overall symmetry is *k*-gonal (prism), tetrahedral, octahedral, or
icosahedral, according to whether the *P* term is D*k*, A4, S4,
or A5, respectively. The *Q* term of D2 or C2, D3 or C3, and D4 or
C4 indicates alignment along a 2-fold, 3-fold, and 4-fold axis, respectively.
The initial *n* term may be followed by A or B if there are two variants.