Can be rephrased as the number of 4x4 Latin squares. Using bruteforce gives 576 configurations.

We complete the Latin square starting with the configuration 1000 2100 3010 4001 filling the zeros, and arrive at four different latin squares 1432 2143 3214 4321 1324 2143 3412 4231 1234 2143 3412 4321 1243 2134 3412 4321 So the answer is $4 \times 4! \times 3! = 576$ as said in the previous post.

A more enlightening way of the bijection to the 4x4 Latin square... Each number in the 4x4 latin square represents an altitude. We are given that the red cubes are once in every altitude, edge, and lateral edge. These Latin squares are once at every altitude, creating the bijection.