Off-shell duality invariance of Schwarzschild perturbation theory

We explore the duality invariance of the Maxwell and linearized Einstein-Hilbert actions on a non-rotating black hole background. On shell these symmetries are electric-magnetic duality and Chandrasekhar duality, respectively. Off shell they lead to conserved quantities; we demonstrate that one of the consequences of these conservation laws is that even- and odd-parity metric perturbations have equal Love numbers. Along the way we derive an action principle for the Fackerell-Ipser equation and Teukolsky-Starobinsky identities in electromagnetism.