We consider the rigidity of origami surfaces with triangular faces whose shapes repeat periodically. We apply a hidden vertex symmetry of the linear rigid- and force-bearing modes, second-order rigidity conditions and numerical validation to characterize the rigid embeddings of such systems. We find generically that even when such embeddings have dihedral angles between faces that are periodic the resulting structure is screw-periodic, with adjacent faces related by discrete translations and rotations. Generically, two-dimensional sets of such structures exist. We also consider the role of the hidden symmetry in non-uniform deformations of such a structure, modifying the associated topological invariant.