One-way intervals of circle maps

An interval in the circle $S^1$ is one-way with respect to a map $f:S^1\to S^1$ if under repeated applications of $f$ all points of the interval move in the same direction. The main result is that every locally one-way interval is either one-way or is the union of two overlapping one-way subintervals. An example is given which illustrates that the latter case can occur; however, it is proved that the latter case cannot occur if the interval is covered by the image of the map. As a corollary, it is shown that if $f$ has periodic points, then every interval which contains no periodic points is either one-way or is the union of two overlapping one-way subintervals.