These magic cubes are bimagic, that is, they are still magic when each cell of them is squared.
order-(2x) bordered diagonal magic cubes
The following magic cubes are probably the first bordered (namely, consecutively concentric) magic cubes of even order in the world, and probably the first bordered diagonal magic cubes in the world.
Each of these cubes is a diagonal magic cube and contains (non-normal) subcubes of all even orders lower than the order of the cube.
Every subcube consists of consecutive integers, and every subcube (except the order-4 one) is a diagonal magic cube.
(An order-4 magic cube cannot be diagonal unless it contains duplicate numbers.)
Note William H. Benson & Oswald Jacobi published the first order-14 diagonal magic cube in 1981. The cube is shown in the following book:
Benson & Jacoby, Magic Cubes New Recreations, Dover,1981, ISBN 0-486-24140-8, pp.116-126.
Note An order-(4x+2) associated diagonal magic cube cannot exist. If such a cube were possible, its oblique squares would be order-(4x+2) (non-normal) associated magic squares with the odd constant. However, such a magic square cannot exist.
Note An order-(4x+2) associated magic hypercube of even dimension cannot exist, but this does not hold for odd dimensions.
Note An order-(4x+2) panmagic hypercube of even dimension cannot exist, but this does not hold for odd dimensions.
In fact, Abhinav Soni made some order-(4x+2) panmagic hypercubes of odd dimension higher than 3 in July 2004.
Open problem Can there exist an order-7 pseudo-pantriagonal diagonal magic cube?
(An order-7 pan-2,3-agonal magic cube cannot exist, so we cannot make order-7 pseudo-pantriagonal diagonal magic cubes by using pan-2,3-agonal magic cubes.)