I think I've got the mystery of the background cracked. They're supposed to be mirrors, reflecting the bottom half on the top and the top half on the bottom. Furthermore, they are conveniently arranged such that any given mirror reflects the background which comes in through a gap perfectly aligned directly across from it. And if we assume the backside of the mirrors are non-reflective, it makes sense that they would cast shadows... granted, they're casting shadows in the sky, but mirrors casting shadows is still more plausible than windows casting shadows.
In order to get the proper colors in each of the squares using Photoshop:
1.) Take a rainbow gradient background without squares, and place in a layer.
2.) Duplicate the layer and flip the duplicate vertically. Place it in the layer behind the original.
3.) Cut out the square pattern from the front layer.
4.) Move the back layer up slightly.
Below is a very, VERY rough draft of the process. There are no shadows and the hol-- I mean, "mirrors" are not lined up properly, but I think it's sufficient to illustrate the concept.