The original cheese was a one-layer affair, you can see a picture at Jaap's Puzzles. The closest classic puzzle to the cheese you see in the picture is the Rubik's UFO, also included on Jaap's pages.
The two layers of this cheese can be rotated with respect to one another and half of the pieces can be turned over along any of the visible cuts. The addition of the extra layer and the coloured stickers on the curved surfaces make this a challenging puzzle to get your head around.
I tend to consider this puzzle solved if the flat faces have only one colour and if each top layer edge matches its bottom layer counterpart. This is how the puzzle was solved for the photograph.