Foldface/bend and preserve UV mapping

I’ve applied a planar mapping to a planar surface. Now I’d like to deform that surface with a sequence of bends and folds whist preserving the UV mapping. Currently bends and folds aren’t preserving that mapping and my texture gets screwed up. Is it possible to deform a surface and preserve its texture mapping?

I [think] I understand now that a planar mapping was the wrong way to go. A planar mapping seems to act like an orthographic projector. Whereas a surface mapping builds UVs based on the structure of the surface control points. When I switched to a surface mapping, I got this. Which is prettier, but still wrong. I guess this means the foldface command isn’t preserving the structure of the surface in a way that generates UVs properly?

(incidentally, both these examples use foldface. When I used bend, it seems to have worked properly. However, performing multiple compound bends with small radii is NOT going to work for me!)

Why is Unwrap not what you want?