Katsuhiko Okamoto's Latch Cube is similar to the standard $3\times 3$ Rubik's cube with the added features that on one of the faces of each of the edge cubies, there is an arrow identifying a direction which the corresponding face may move.
If a face has edges with arrow(s) all pointing in the same direction, the face can rotate in that direction; if there are no arrows on the face, the face can rotate in both directions; and if there are arrows pointing in opposite directions, then that face cannot rotate.
Clearly these features mean that, unlike the Rubik's cube, the adjacency matrix/"Cayley graph" of the Latch Cube is directed, and if we don't consider self-loops, the Latch Cube is irregular. I think there may be positions wherein the only move available is a one-way rotation of one of the faces, because the other faces are locked.
I'm curious now about what can be said about the stationary distribution of the Latch Cube. Famously the stationary distribution of the Rubik's cube is uniform over the $43,252,003,274,489,856,000$ positions - after mixing the cube for long enough, each of the above positions is equally likely to occur.
For example, if, starting from the solved state, we uniformly pick at random one of the quarter-turns available, what is the expected number of moves before we are at the solved position again?
Unlike the Rubik's cube, the solved position of the Latch Cube seems awfully special...