  It is well known that asymptotically flat black holes in general relativity
have vanishing tidal Love numbers. In the case of Schwarzschild and Kerr black
holes, this property has been shown to be a consequence of a hidden structure
of ladder symmetries for the perturbations. In this work, we extend the ladder
symmetries to non-rotating charged black holes in general relativity. As
opposed to previous works in this context, we adopt a more general definition
of Love numbers, including quadratic operators that mix gravitational and
electromagnetic perturbations in the point-particle effective field theory. We
show that the calculation of a subset of those couplings in full general
relativity is affected by an ambiguity in the split between source and
response, which we resolve through an analytic continuation. As a result, we
derive a novel master equation that unifies scalar, electromagnetic and
gravitational perturbations around Reissner--Nordstr\"om black holes. The
equation is hypergeometric and can be obtained from previous formulations via
nontrivial field redefinitions, which allow to systematically remove some of
the singularities and make the presence of the ladder symmetries more manifest.
