Simple flat twists are ubiquitous within the world of origami tessellations, and they make an excellent introduction to the mathematics of flat-foldable multi-vertex origami. Many flat-folded tessellations display several common features. They are often composed of a grid of creases in a regular array, and the folded form is similarly regular and periodic—that is, it repeats in two different directions. Flat-foldability and parallelness of pleats enforce the commonality of the twist angle for all pleats. Any polygonal twists can be labeled by the type of crease encountered as one goes sequentially around the central polygon. Irregular polygonal twists, like their regular counterparts, exist in cyclic form, where the central polygon is outlined by all mountain or all valley folds, as well as versions in which some of the surrounding creases and their attached pleats are inverted.