This puzzle is a fully-functional Siamese 2x2x2 magic cube. being fully-functional means that the two fused cubes can exchange pieces with each other since the upper and lower layers rotate on a single axis through the center of the two cubes. The puzzle is essentially a shape-modification of my Fun Block puzzle (or that puzzle is a shape-modification of this one).
The Fully Functional Siamese 2x2x2 has been designed to use standard sized 3x3x3 stickers, which can be purchased from a variety of sources.
I recommend that you purchase the puzzle in "Black Strong & Flexible". It costs a bit more than the white material, but it's worth it since the finished puzzle will look and work better. All of the parts snap together, so no additional screws, springs, or other hardware are required. Be sure to test the assembly before applying any lubricant to the puzzle, as it will likely be unnecessary.
With the Fully Functional Siamese 2x2x2 you can't make a mistake with the assembly. Since the parts can be assembled onto the core even after the core halves are snapped together, you can assemble the puzzle in any order you desire. Of course, this means the puzzle is also very easy to take apart again.