Combinatorial geometry; Graph drawing; Planar graphs; Untangling problem

C. Thomassen classified Klein-bottle polyhexes into five classes [C. Thomassen, Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface, Trans. Amer. Math. Soc. 323 (1991) 605–635]. In this paper, by implementing cutting and gluing operations, we reclassify the five classes of Klein-bottle polyhexes into two classes—the bi