Rigidly foldable tubes, initially derived from mirroring and interconnecting corresponding edges of Miura-ori cells, possess bi-directional flat-foldability and exhibit out-of-plane stiffness due to their one-degree-of-freedom kinematic movement in their regular configuration. Various methods have been established for enhancing their structural complexity and stiffness, including edge and face connections, as well as zipping and interleaving, leading to the creation of advanced structures like foldable sandwiches and metamaterials. In this study, we introduce a novel family of cellular structures where orthogonally aligned rigid-foldable tubes are in a woven configuration. We show a computational process to achieve the tubular designs following a translational surface for any given weaving pattern (including existing interleaved tubes) by introducing the idea of flip-flop joints that can switch the up-and-down relation between contact tubes by changing their orientation. This research not only expands the repertoire of origami-based engineering techniques but also offers a practical toolset for designers and engineers seeking innovative solutions in structural optimization.