  The Grassmannian formulation of $\mathcal{N}=4$ super Yang-Mills theory
expresses tree-level scattering amplitudes as linear combinations of residues
from certain contour integrals. BCFW bridge decompositions using adjacent
transpositions simplify the evaluation of individual residues, but orientation
information is lost in the process. We present a straightforward algorithm to
compute relative orientations between the resulting coordinate charts, and we
show how to generalize the technique for charts corresponding to sequences of
any not-necessarily-adjacent transpositions. As applications of these results,
we demonstrate the existence of a signed boundary operator that manifestly
squares to zero and prove via our algorithm that any residues appearing in the
tree amplitude sum are decorated with appropriate signs so all non-local poles
cancel exactly, not just mod 2 as in previous works.
