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Lattices and their underlying symmetries play a central role in determining the physical properties and applications of many natural and engineered materials. This research offers a comprehensive solution to a long-standing challenge regarding the lattice-based materials: how to systematically construct a lattice and transform it among different symmetric configurations in a predictable, scalable, and reversible way. By bridging the lattice geometry and rigid-folding kinematics of a group of origamis consisting of generic degree-4 vertices, we successfully construct all types of two- and three-dimensional Bravais lattices, and demonstrate that they can undergo all diffusionless phase transformations via rigid folding (i.e., dilation, extension, contraction, shear, and shuffle). Such folding-induced lattice transformations can trigger fundamental lattice-symmetry switches, which can either maintain or reconstruct the nearest-neighbor relationships according to a continuous symmetry measure. This study can foster the next generation of transformable lattice structures and materials with on-demand property tuning capabilities.