We give a mathematical treatment to the concept of ordinal equivalence defined relative to all m! possible permutations of parallel axes. We prove that the ordinal equivalence is determined by the pair-wise co-monotonicity equivalence relations, thus leading to simple algorithmic procedures for finding the corresponding partition. Each ordinal equivalence class can be visualized as a profile of co-monotone polylines, in this way preventing any clutter at the image. We illustrate our approach with two datasets taken from the literature.