3x3x3 Study - picture cubes, 3x3x3 variants

There are quite a lot of 3x3x3 variant puzzles out there for which your typical Fridrich method will not quite get your there!
I'm talking of picture cubes, Shepard cubes, Pochmann Cubes, Master Pyramorphinxes, etc, etc.
My approach here is to study my algorithms and see exactly what effects we get: for example, a classic Sune appears to orient 3 corners but it also permutes 3 already oriented edges. Which ones? Well, if you label the pieces with little temporary Post-It note stickers you can see exactly what happens.
 
Here's a nice algorithm: U (R L U2 R' L') U (R L U2 R' L')
Watch what happens: it rotates the U centre by 90 degrees! It's quite easy to remember too.