In woven textile-form design, layers may be stacked in a pleat structure, in order to provide greater area from a limited weaving width, or to create shaping in a rectangular cloth. The layers are typically joined by weaving them together where their edges meet. However, this method of joining creates a thickened seam, and may prevent the layers from opening flat. This paper presents a woven textile hinge structure, which enables joined layers to be opened flat, without adding extra thickness, or requiring further finishing. Its utility is demonstrated through the Twistbox, an eight-layer textile-form, woven flat, which unfolds into a cube, and can be collapsed and unfolded indefinitely. The structure was produced through experimental design research, in the context of a bio-inspired design collaboration. Inspired by natural suture structures, the hinge structure broadens possibilities for woven textile-form design, while its development provides a case for how biomimetics may be applied in textile design.