Slice 7x7x7 is a variation to the Rubik's Cube, in which layer 1 is rigidly connected to layer 7, 2 to 6, and 3 to 5. The opposite-layer connection is there in all three directions. The result is a puzzle that only makes nice symmetrical patterns, no matter how you scramble it.
The creation of this puzzle was triggered by email communication with Nersés Aramian. He suggested to build a puzzle based on the projective plane, i.e. pairs of pieces opposite from the origin always remain opposite. Slice 7x7x7 satisfies Nersés' requirement, although this puzzle may not be what he had in mind.