  Unique Sink Orientations (USOs) of cubes can be used to capture the
combinatorial structure of many essential algebraic and geometric problems. For
various structural and algorithmic questions, including enumeration of USOs and
algorithm analysis, it is crucial to have systematic constructions of USOs.
While some construction methods for USOs already exist, each one of them has
some significant downside. Most of the construction methods have limited
expressivity -- USOs with some desired properties cannot be constructed. In
contrast, the phase flips of Schurr can construct all USOs, but the operation
is not well understood. We were inspired by techniques from cube tilings of
space; we expand upon existing techniques in the area to develop generalized
rewriting rules for USOs. These rewriting rules are a new construction
framework which can be applied to all USOs. The rewriting rules can generate
every USO using only USOs of lower dimension. The effect of any specific
rewriting rule on an USO is simple to understand. A special case of our
construction produces a new elementary transformation of USOs, which we call a
partial swap. We further investigate the relationship between partial swaps and
phase flips and generalize partial swaps to phase swaps.
