  The starting point of this work was an intriguing similarity between the
behaviour of fields near a degenerate horizon and near the infinity of an
asymptotically flat spacetime, as revealed by the scattering theory for Dirac
fields in the ``exterior'' region of the extreme Kerr - de Sitter black hole,
developed by one of the authors (JB). However, in that situation, the
comparison was somewhat clouded by some of the analytical techniques used in
intermediate steps of the proof. The aim of the present work is to clarify the
comparison further by studying instead the peeling behaviour of solutions to
the wave equation at an extremal horizon. We focus first on the extreme
Reissner-Nordstr\"om black hole, for which the Couch-Torrence inversion (a
global conformal isometry that exchanges the horizon and infinity) makes the
analogy explicit. Then, we explore more general spherically symmetric
situations using the Couch-Torrence inversion outside of its natural context.
