I'm a competitive speedcuber (Rubik's cube competition average 18.74 sec.) and thus have most of the standard competition twisty puzzles.

So I asked for a twisty puzzle that would surprise me... and boy did my Santa come through.

Edit: SOLVED!

It turns out that

1. You can only turn a face 180º; this makes solving the corners fairly easy.

2. Blindly turning the right face over and over again will automatically orient the edges in the vertical center slice. So once you have the edges in the right places, just twisting over and over again will solve the cube.

Getting the edges in the right places is not all that intuitive, though, so I'll be working on standardizing that now.

Thanks again to my Santa for an amazing gift!