  The extreme Reissner-Nordstr\"om solution has a discrete conformal isometry
that maps the future event horizon to future null infinity and vice versa, the
Couch-Torrence inversion isometry. We study the dynamics of a probe Maxwell
field on the extreme Reissner-Nordstr\"om solution in light of this symmetry.
We present a gauge fixing that is compatible with the inversion symmetry. The
gauge fixing allows us to relate the gauge parameter at the future horizon to
future null infinity, which further allows us to study global charges for large
gauge symmetries in the exterior of the extreme Reissner-Nordstr\"om black
hole. Along the way, we construct Newman-Penrose and Aretakis like conserved
quantities along future null infinity and the future event horizon,
respectively, and relate them via the Couch-Torrence inversion symmetry.
