Tidal effects have long ago locked the Moon in a synchronous rotation with the Earth and progressively increase the Earth-Moon distance. This 'tidal acceleration' hinges on dissipation. Binaries containing black holes may also be tidally accelerated, dissipation being caused by the event horizon-a flexible, viscous one-way membrane. In fact, this process is known for many years under a different guise: superradiance. Here, we provide compelling evidence for a strong connection between tidal acceleration and superradiant scattering around spinning black holes. In general relativity, tidal acceleration is obscured by the gravitational-wave emission. However, when coupling to light scalar degrees of freedom is allowed, an induced dipole moment produces a 'polarization acceleration', which might be orders of magnitude stronger than tidal quadrupolar effects. Consequences for optical and gravitational-wave observations are intriguing and it is not impossible that imprints of such a mechanism have already been observed.