  Black holes can be electromagnetically charged, or carry vector charge from
new fundamental fields. Their response to small fluctuations is of paramount
importance to study gravitational wave generation. However, the usual even and
odd sectors of gravitoelectromagnetic waves couple if the black hole is
magnetically charged, a fact that complicates significantly the perturbative
approach. In this paper, perturbation theory based on harmonic expansion is
extended to have manifest invariance under electric-magnetic duality. As a
result, the equations decouple into two generalised even and odd sectors, each
governed by master wave equations that include the most general coupling to a
dyonic source. These can be used to compute, in a simple manner, the
gravitational and electromagnetic radiation emitted in the interaction of the
most general spherically symmetric black holes of the Einstein--Maxwell theory
with electromagnetically charged matter.
