The Winding Conjecture for fully flipped interval exchange maps

Let $\CETn$ denote the family of maps used in the source, and consider periodic trajectories of a map $F\in\CETn$. Winding Conjecture. The winding number of a periodic trajectory does not depend on the trajectory and defines an invariant $\operatorname{wind}(F)$ of the map. Moreover,

$$\operatorname{wind}(F)=\pm 2n \quad\text{if } n \text{ is odd},$$

and

$$\operatorname{wind}(F)=\pm n \quad\text{if } n \text{ is even}.$$

In the tiling-billiard interpretation, this says that periodic orbits obtained by the same scissor cut have the same orientation with respect to infinity. The preceding lemma establishes related winding facts for reflection systems, but the conjecture itself is not resolved in the supplied text.