  We examine certain symmetries in the deficiencies of a rational surgery on a
knot in $S^3$ by comparing the $\text{Spin}^c$-structures on the rational
surgery with those on a related integral surgery. We then provide an
application of these symmetries in the form of a theorem that obstructs Dehn
surgeries in $S^3$. This last part unifies and generalises theorems by Greene
underlying his work on the cabling conjecture, the lens space realisation
problem, and the unknotting number of alternating 3-braids.
