  Black hole spectroscopy is a powerful tool to probe the Kerr nature of
astrophysical compact objects and their environment. The observation of
multiple ringdown modes in gravitational waveforms could soon lead to
high-precision gravitational spectroscopy, so it is critical to understand if
the quasinormal mode spectrum is stable against perturbations. It was recently
shown that the pseudospectrum can shed light on the spectral stability of black
hole quasinormal modes. We study the pseudospectrum of Reissner-Nordstr\"om
spacetimes and we find a spectral instability of scalar and gravitoelectric
quasinormal modes in subextremal and extremal black holes, extending similar
findings for the Schwarzschild spacetime. The asymptotic structure of
pseudospectral contour levels is the same for scalar and gravitoelectric
perturbations. By making different gauge choices in the hyperboloidal slicing
of the spacetime, we find that the broad features of the pseudospectra are
remarkably gauge-independent. The gravitational-led and electromagnetic-led
quasinormal modes of extremal Reissner-Nordstr\"om black holes exhibit "strong"
isospectrality: not only their spectrum coincides, but the whole pseudospectrum
is the same for both classes of perturbations. We observe that a conformal
duality between the extremal horizon and spacetime boundaries at infinity is
responsible for such "strong" isospectrality property.
