Take a cube(3x3x3) and re-color it in two colors, as follows:
Make 2 opposite faces with color 1 in the center, and color 2 on the
rest. Call them "dots". Take another pair of opposite faces and make
the 4 corners color 2 and the cross color 1. Call them "pluses".
Finally take the last pair and make an "H" with color 1 and the 2
edges NOT adjacent to "dots" color 2. Call them "H's". In this cube,
every edge is the same, so is each corner, and the centers. The result
is a cube where you only have to solve orientation problems, and never
need to position any cubie. However, it's not quite the entire orientation
sub-cube - if you flip all 12 edges, you can't tell the difference. Can
anybody come up with a coloring (2 or 3 colors - the centers could all be
colored with a different color from the edges and corners. In fact, I guess
you could use 6 colors; the only necessity is for all edges to be the same,
and all corners the same. )... come up with a coloring which uses the
complete twisting sub-group.
By the way, an elegant solution to the "edge-only" cube is to recolor
an Octahedron Cube so it is vertex centered. Much nicer than peeling all
the corners of a regular cube.