This isn't strictly speaking twisty, but probably of interest to those on here. There's a construction set made by hoberman
in which you put together squares, triangles, and lines, like with other contruction toys, but with the added twist that everything you make with it can be expanded and collapsed. I think these sets are unbelievably cool, but sadly the general public doesn't seem to agree because they haven't sold well and the line has been discontinued.
On the plus side, because the line is discontinued the most excellent store math 'n' stuff
located here in Seattle is now selling all their expandagon merchandise at half off. I highly recommend snapping it up cheap before it's all gone.