  Uniqueness results for asymptotically locally flat and asymptotically flat
$S^1$-symmetric gravitational instantons are proved using a divergence identity
of the type used in uniqueness proofs for static black holes, combined with
results derived from the $G$-signature theorem. Our results include a proof of
the $S^1$-symmetric version of the Euclidean Black Hole Uniqueness conjecture,
a uniqueness result for the Taub-bolt family of instantons, as well as a proof
that an ALF $S^1$-symmetric instanton with the topology of the Chen-Teo family
of instantons is Hermitian.
