You can view 1,2,3,4-tetrasomethingcyclobutane as a square with black and/or white corners (black representing e.g. up, then white representing down configuration of the substituent). Two squares represent the same object when after applying some of the following operations on the first you get the second one:
- rotation in the paper plane (of course)
- mirroring AND color inversion
By drawing all 24 = 16 possible squares and using these two rules, you quickly find that they represent only 4 different objects, as seen on the following sketch.
Note that the second "controversial" operation mirroring AND color inversion is in our scenario, because of D2 symmetry of all squares, equivalent with more proper, "mirror-free" operation - rotation around some in-plane axis, considering that obverse of the square has inverse colors. (Because of the symmetry, the operation can even be reduced to color inversion only.)