Origami tessellations are often composed of the intersections of flat pleats. In this style, there are an infinite number of possible twist systems. This paper presents a notation system for uniquely describing these systems on a triangle grid. After providing the notation structure, it outlines pleat operations and how the notation describes them. The paper also identifies sets of pleat systems that have common properties, and analyzes how applying the above operations affects those properties. This delineation will provide readers with useful techniques for analyzing twists in a tiling context, or as an isolated system of folds.