Self-similar Magic Cubes

Cube_4-Trump-1   Center symmetrical cubes         Associated  

 1  63  62   4    48  18  19  45    32  34  35  29    49  15  14  52
60   6   7  57    21  43  42  24    37  27  26  40    12  54  55   9
56  10  11  53    25  39  38  28    41  23  22  44     8  58  59   5
13  51  50  16    36  30  31  33    20  46  47  17    61   3   2  64    

This cube is symmetric around a point in the center of the cube.
It is the only type of symmetry that appears in odd order cubes (or squares). This type of cube is called associated, or quite often it is called symmetric. However, there is one other type of symmetry that can appear in even order magic squares. There are three other types of even order symmetrical cubes.

This is a disguised version of the Schubert cube of 1898

Why only center symmetry for odd order cubes (and squares). A later comment from Walter.
Things ae really different with odd orders.
Only the center symmetry is possible in that case.
All axis would contain more then one cell,
and there is only ONE number that is complementary to itself.

Cube_4-Trump-2   Orthogonal axis symmetrical cubes    Not associated

 1  24  62  43    16  25  51  38    61  44   7  18    52  37  10  31
48  57  19   6    33  56  30  11    20   5  42  63    29  12  39  50
59  46   8  17    54  35   9  32     2  23  60  45    15  26  53  36
22   3  41  64    27  14  40  49    47  58  21   4    34  55  28  13

This cube is symmetrical around an axis line parallel to four of the cube faces.

The four horizontal squares are associative.
This type of symmetry works only on even order magic cubes (or squares).

This simple magic cube has a bonus feature (not associated with symmetry).
 The four horizontal planes and two of the six oblique planes are simple magic squares.

Cube_4-Trump-3   Diagonal axis symmetrical cubes     not associated

 1   4  62  63    23  31  44  32    42  40  18  30    64  55   6   5
10  58  45  17    25  53  36  16    34  12  38  46    61   7  11  51
59  54   8   9    47  27  13  43    21  29  52  28     3  20  57  50
60  14  15  41    35  19  37  39    33  49  22  26     2  48  56  24   

This cube is symmetrical around a diagonal axis.

See further comments and illustration in the next section.

Cube_4-Trump-4    Plane symmetrical cubes      Not associated

 1   5  61  63    40  15  44  31    25  50  21  34    64  60   4   2
14  58  52   6    18  55  27  30    47  10  38  35    51   7  13  59
62  43   9  16    46  37  11  36    19  28  54  29     3  22  56  49
53  24   8  45    26  23  48  33    39  42  17  32    12  41  57  20    

There are billions of cubes with this symmetry.
This is an example of a cube that is symmetrical across an orthogonal plane.

Note:

There exists another geometrical symmetry of cubes. But 16 cells are positioned directly in such a symmetry plane.

The numbers in those cells would be their own complements.

Cubes of odd order have got only one number with this property, cubes of even order none.

Thus there are no magic cubes that are symmetrical with respect to such planes.

For example: 62 and 3 is a pair, 63 and 2 is another pair. The axis is the dividing line midway between all the pairs.

If the two numbers of each complementary pair in the cube are exchanged, a different aspect of the same cube will be obtained. This is true for any cube that has one of these four types of symmetry.

That is why Mutsumi Suzuki coined the term 'self-similar'.
Of course, if the cube is not symmetrical, exchanging each number with its complement will result in a completely different cube!