I got a novelty cube at as swag at a convention once; it was like a normal cube, but with slogans written on the squares of the cube. If you solved the cube, you could read the slogan.
This got me to wondering if it's possible to solve this cube such that some of the words faced the wrong way.
If you have a standard Rubik's cube, and draw arrows on each label (all arrows on each face point in the same direction), is it possible to solve the cube such that some of the arrows face different directions from the original orientation? If so, how many distinct solved cubes are there?
Note: obviously, for all the non-center faces, there is only one possible orientation. For example, if the red-blue edge has "up" on the blue face pointing at red, then the red-blue-yellow and red-blue-green corners must be next to that edge, and must also have "up" on the blue faces point toward red. Thus, the orientations of the edges and corners is fixed, but it might be possible for the centers to end up rotated.