Twist structures in origami tessellations can be discretized and placed on a number line with a fixed transformation between each twist. These number lines represent not only potential for infinite expansion of twists, but also the connection between twists on opposite sides of the paper. When extended symmetries are applied to tessellation tilings these number lines of twists provide relative positions that enable the construction of design equations representing the pleat shifts needed to produce the given symmetry. These extended symmetries enable the construction of ever-more elaborate and numerous origami tessellation designs while using only the simplest twists and folding methods.