  A unique sink orientation (USO) is an orientation of the $n$-dimensional
hypercube graph such that every non-empty face contains a unique sink. Schurr
showed that given any $n$-dimensional USO and any dimension $i$, the set of
edges $E_i$ in that dimension can be decomposed into equivalence classes
(so-called phases), such that flipping the orientation of a subset $S$ of $E_i$
yields another USO if and only if $S$ is a union of a set of these phases. In
this paper we prove various results on the structure of phases. Using these
results, we show that all phases can be computed in $O(3^n)$ time,
significantly improving upon the previously known $O(4^n)$ trivial algorithm.
Furthermore, we show that given a boolean circuit of size $poly(n)$ succinctly
encoding an $n$-dimensional (acyclic) USO, it is PSPACE-complete to determine
whether two given edges are in the same phase. The problem is thus equally
difficult as determining whether the hypercube orientation encoded by a given
circuit is an acyclic USO [G\"artner and Thomas, STACS'15].
