Results following @MaartenHavinga's corrected answer for n=4:

Group 0: 
 [[0 0 0]]
Group 1: 
 [[0 0 1]
 [0 1 0]
 [1 0 0]]
Group 2: 
 [[0 0 2]
 [0 2 0]
 [2 0 0]]
Group 3: 
 [[0 0 3]
 [0 3 0]
 [3 0 0]]
Group 4: 
 [[0 1 1]
 [1 0 1]
 [1 1 0]]
Group 5: 
 [[0 1 2]
 [1 2 0]
 [2 0 1]]
Group 6: 
 [[0 2 1]
 [1 0 2]
 [2 1 0]]
Group 7: 
 [[0 1 3]
 [0 3 1]
 [1 0 3]
 [1 3 0]
 [3 0 1]
 [3 1 0]]
Group 8: 
 [[0 2 2]
 [2 0 2]
 [2 2 0]]
Group 9: 
 [[0 2 3]
 [0 3 2]
 [2 0 3]
 [2 3 0]
 [3 0 2]
 [3 2 0]]
Group 10: 
 [[0 3 3]
 [3 0 3]
 [3 3 0]]
Group 11: 
 [[1 1 1]]
Group 12: 
 [[1 1 2]
 [1 2 1]
 [2 1 1]]
Group 13: 
 [[1 1 3]
 [1 3 1]
 [3 1 1]]
Group 14: 
 [[1 2 2]
 [2 1 2]
 [2 2 1]]
Group 15: 
 [[1 2 3]
 [1 3 2]
 [2 1 3]
 [2 3 1]
 [3 1 2]
 [3 2 1]]
Group 16: 
 [[1 3 3]
 [3 1 3]
 [3 3 1]]
Group 17: 
 [[2 2 2]]
Group 18: 
 [[2 2 3]
 [2 3 2]
 [3 2 2]]
Group 19: 
 [[2 3 3]
 [3 2 3]
 [3 3 2]]
Group 20: 
 [[3 3 3]]

array([[[ 1.62, -0.61, -0.53, -1.07],
        [-0.61,  0.87, -2.3 , -0.76],
        [-0.53,  1.74,  0.32, -0.25],
        [-1.07, -0.76, -0.25,  1.46]],

       [[-0.61,  0.87,  1.74, -0.76],
        [ 0.87, -2.06, -0.32, -0.38],
        [-2.3 , -0.32,  1.13, -1.1 ],
        [-0.76, -0.38, -1.1 , -0.17]],

       [[-0.53, -2.3 ,  0.32, -0.25],
        [ 1.74, -0.32,  1.13, -1.1 ],
        [ 0.32,  1.13, -0.88,  0.04],
        [-0.25, -1.1 ,  0.04,  0.58]],

       [[-1.07, -0.76, -0.25,  1.46],
        [-0.76, -0.38, -1.1 , -0.17],
        [-0.25, -1.1 ,  0.04,  0.58],
        [ 1.46, -0.17,  0.58, -1.1 ]]])

Results following @MaartenHavinga:

I attempted to follow Maarten's instructions to generate the $(0,0,0)$ corner of the cube (the first dimension is the row, second is the column, third is the depth, i.e. the "upper left front" part of the cube).

When $n=3$, the results are good:

Group 0: 
 [[0 0 0]]
Group 1: 
 [[0 0 1]
 [0 1 0]
 [1 0 0]]
Group 2: 
 [[0 0 2]
 [0 2 0]
 [2 0 0]]
Group 3: 
 [[0 1 1]
 [1 0 1]
 [1 1 0]]
Group 4: 
 [[0 1 2]
 [0 2 1]
 [1 0 2]
 [1 2 0]
 [2 0 1]
 [2 1 0]]
Group 5: 
 [[0 2 2]
 [2 0 2]
 [2 2 0]]
Group 6: 
 [[1 1 1]]
Group 7: 
 [[1 1 2]
 [1 2 1]
 [2 1 1]]
Group 8: 
 [[1 2 2]
 [2 1 2]
 [2 2 1]]
Group 9: 
 [[2 2 2]]

array([[[ 1.62, -0.61, -0.53],
        [-0.61, -1.07,  0.87],
        [-0.53,  0.87, -2.3 ]],

       [[-0.61, -1.07,  0.87],
        [-1.07,  1.74, -0.76],
        [ 0.87, -0.76,  0.32]],

       [[-0.53,  0.87, -2.3 ],
        [ 0.87, -0.76,  0.32],
        [-2.3 ,  0.32, -0.25]]])

However, when $n=4$, the results are slightly off:

Group 0: 
 [[0 0 0]]
Group 1: 
 [[0 0 1]
 [0 1 0]
 [1 0 0]]
Group 2: 
 [[0 0 2]
 [0 2 0]
 [2 0 0]]
Group 3: 
 [[0 0 3]
 [0 3 0]
 [3 0 0]]
Group 4: 
 [[0 1 1]
 [1 0 1]
 [1 1 0]]
Group 5: 
 [[0 1 2]
 [1 2 0]
 [2 0 1]]
Group 6: 
 [[0 2 1]
 [1 0 2]
 [2 1 0]]
Group 7: 
 [[0 1 3]
 [0 3 1]
 [1 0 3]
 [1 3 0]
 [3 0 1]
 [3 1 0]]
Group 8: 
 [[0 2 2]
 [2 0 2]
 [2 2 0]]
Group 9: 
 [[0 2 3]
 [0 3 2]
 [2 0 3]
 [2 3 0]
 [3 0 2]
 [3 2 0]]
Group 10: 
 [[0 3 3]
 [3 0 3]
 [3 3 0]]
Group 11: 
 [[1 1 1]]
Group 12: 
 [[1 1 2]
 [1 2 1]
 [2 1 1]]
Group 13: 
 [[1 1 3]
 [1 3 1]
 [3 1 1]]
Group 14: 
 [[1 2 2]
 [2 1 2]
 [2 2 1]]
Group 15: 
 [[1 2 3]
 [2 3 1]
 [3 1 2]]
Group 16: 
 [[1 3 2]
 [2 1 3]
 [3 2 1]]
Group 17: 
 [[1 3 3]
 [3 1 3]
 [3 3 1]]
Group 18: 
 [[2 2 2]]
Group 19: 
 [[2 2 3]
 [2 3 2]
 [3 2 2]]
Group 20: 
 [[2 3 3]
 [3 2 3]
 [3 3 2]]
Group 21: 
 [[3 3 3]]

array([[[ 1.62, -0.61, -0.53, -1.07],
        [-0.61,  0.87, -2.3 , -0.76],
        [-0.53,  1.74,  0.32, -0.25],
        [-1.07, -0.76, -0.25,  1.46]],

       [[-0.61,  0.87,  1.74, -0.76],
        [ 0.87, -2.06, -0.32, -0.38],
        [-2.3 , -0.32,  1.13, -1.1 ],
        [-0.76, -0.38, -0.17, -0.88]],

       [[-0.53, -2.3 ,  0.32, -0.25],
        [ 1.74, -0.32,  1.13, -0.17],
        [ 0.32,  1.13,  0.04,  0.58],
        [-0.25, -1.1 ,  0.58, -1.1 ]],

       [[-1.07, -0.76, -0.25,  1.46],
        [-0.76, -0.38, -1.1 , -0.88],
        [-0.25, -0.17,  0.58, -1.1 ],
        [ 1.46, -0.88, -1.1 ,  1.14]]])

I will refer to positions starting from index 1. In the 2nd depth position, the entry in the 3rd row and 4th column (-1.1) is distinct from the entry in the 4th row and 3rd column (-0.17). This is because the indices are (3, 4, 2) and (4, 3, 2) which are an odd permutation.

However, in order for the cube to be rotation invariant to rotations about the row and column axes (i.e., around the entry -0.88 in the second depth dimension), these two entries should be equal.

I have a cube $X\in \mathbb R^{N\times N\times N}$ such that no matter how the cube is rotated by $90^\circ$ along any of the axes, the result is unchanged. What is the maximum number of unique entries in the cube if $N$ is odd?

Here is my answer. I am sure that it is less than or equal to the true answer.

Let $n=(N+1)/2$. There exists a subcube of shape $\mathbb R^{n\times n\times n}$ that contains all of the unique entries. This subcube must be on the corner of the cube, so I can imagine the indices of the entries as being in $\mathbb N_n^3,\mathbb N_n:=1,...,n$.

The claim I am unsure of: the indices of any one of the unique entries are the same up to a permutation, e.g., the $(1,2,3), (1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)$ entries must all be the same. This is sufficient for the cube to be rotation invariant, but it may not be necessary, which would allow for the final answer to be greater.

Assuming that this claim is true, it's enough to just count the number of sets of indices that are equal up to a permutation. That is the same as unordered sampling with replacement, and thus, the number of unique entries is $\binom{n+3-1}{3} = \binom{n+2}{3}$

I have a cube $X\in \mathbb R^{N\times N\times N}$ such that no matter how the cube is rotated by $90^\circ$ along any of the axes, the result is unchanged. What is the maximum number of distinct entries in the cube if $N$ is odd?

Here is my answer. I am sure that it is less than or equal to the true answer.

Let $n=(N+1)/2$. There exists a subcube of shape $\mathbb R^{n\times n\times n}$ that contains all of the distinct entries. This subcube must be on the corner of the cube, so I can imagine the indices of the entries as being in $\mathbb N_n^3,\mathbb N_n:=1,...,n$.

The claim I am unsure of: the indices of any one of the distinct entries are the same up to a permutation, e.g., the $(1,2,3), (1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1)$ entries must all be the same. This is sufficient for the cube to be rotation invariant, but it may not be necessary, which would allow for the final answer to be greater.

Assuming that this claim is true, it's enough to just count the number of sets of indices that are equal up to a permutation. That is the same as unordered sampling with replacement, and thus, the number of distinct entries is $\binom{n+3-1}{3} = \binom{n+2}{3}$