Alright, I've had the time to take a better look at Veronika's diagrams. They're really well-done, I feel like I finally understand what's going on! (quite a feat considering the headaches I've amassed from trying to figure out what's happening from several other sites)

It's easier to see how handles would work thanks to those. Here's an example toggling between P1, P2, PM and PMM.

http://www.freeimagehosting.net/kouh5

(hopefully you can tell the pair, symmetry and rotation handles)

With an additional handle for glide reflection, it should be possible to connect all shapes with rectangular base tiles. I'm not sure how to fit in P3 and P4 though.

What's more, it may be possible to convert between most of the triangle tiles by clicking on a corner, and dragging around to choose the number of tiles in that corner (3, 4, 6, 8 or 12). The shape of the tile would have to be updated automatically.

The problem is that the users may then be tempted to drag other handles around to convert between pair, symmetry and rotate to play around, but the resulting output may not correspond to a tillable symmetry. There should be a method of warning in that case. Also, to make things less confusing, it may be possible to force one corner to be a "primary rotation corner" (the corner with the largest amount of tiles touching it) and a "secondary rotation corner."

Does anyone feel up to trying to make a diagram with all 17 tilings as well as conversion methods between them? If you somehow manage to do that, I'll try to make a new tiling GUI proposal around it.