The “Bendy Ply” I have used is always a three ply construct with a thin & stiff centre layer, (1mm or less), which has its grain at 90 deg to the 2 outer layers. The outer layers are a a very soft straight grained timber. The grain on the concave side compresses whilst that on the convex side opens out slightly, the “spine allowing it to bend without snapping.
Two pieces will keep their shape if glued together. It bends very easily, but not compound curves of course.
It is available in, “Long Grain” or “Cross Grain”, depending on whether you want an 8ft high tube or a 4ft high tube!
You can also buy flexible MDF, basically a sheet of MDF with saw kerfs machined into it, it snaps easily, and like most kerfed timber it ends up being multi faceted, and needs sanding smooth if visible. “Bendy Ply” is much better in my experience.
If you want to make your own ply, then “constructional” veneer is available up to about 3mm thick, normal veneer is only 0.6mm. It might only be around 6 inches wide though.
Mark.