We introduce a general geometric framework for the construction of polyhedra and polyhedral complexes that are bifoldable, i.e. bi-directional, flat-foldable into two orthogonal planes. This vastly generalizes origami folds known as the Miura pattern, the origami tube and the Eggbox pattern. Our polyhedra are generalized zonohedra based on 1-parameter family stars of vectors in \(\mathbb {R}^3 \) that deform in specific ways while the polyhedra and polyhedral complexes are flat-folded. After describing the framework, its basic features, and the general design process, we give several new examples of infinite doubly periodic, triply periodic, and fractal bifoldable polyhedra.