In 1917, Huntington and Kline, followed by Huntington in 1924, studied systems of axioms for ternary relations aiming to capture the concepts of linear order (called betwenness) and cycle order, respectively. Among many other properties, they proved there are several independent set of axioms defining either linear order or cycle order. In this work, we consider systems arising from the four common axioms of linear order and cycle order, together with a new axiom F, stating that if a tuple abc is in the relation, then either cba or bca is in the relation as well. Although, at first glace this allows for a much richer type of systems, we prove that these are either of linear order or cycle order type. Our main result is a complete characterization of the finite systems satisfying the set of axioms {B, C, D, F, 2}, where B, C, D and 2 are axioms presented by Huntington. Unlike what happens in the previous situation, with this modification we obtain a larger family of systems which we characterize in terms of digraphs.