Oriented classification of lens spaces

With a consistent surgery orientation convention, $L(p,q)$ and $L(p,q^{\prime })$ are orientation-preservingly related by a homeomorphism exactly when $q^{\prime }\equiv q^{\pm 1}(\mathrm{mod}p)$. An orientation-reversing homeomorphism instead requires $q^{\prime }\equiv -q^{\pm 1}(\mathrm{mod}p)$.