  We characterise the geometrical nature of smooth supertranslations defined on
a generic non-expanding horizon (NEH) embedded in vacuum. To this end we
consider the constraints imposed by the vacuum Einstein's equations on the NEH
structure, and discuss the transformation properties of their solutions under
supertranslations. We present a freely specifiable data set which is both
necessary and sufficient to reconstruct the full horizon geometry, and is
composed of objects which are invariant under supertranslations. We conclude
that smooth supertranslations do not transform the geometry of the NEH, and
that they should be regarded as pure gauge. Our results apply both to
stationary, and non-stationary states of a NEH, the later ones being able to
describe radiative processes taking place on the horizon. As a consistency
check we repeat the analysis for BMS supertranslations defined on null
infinity, $\mathcal{I}$. Using the same framework as for the NEH we recover the
well known result that BMS supertranslations act non-trivially on the free data
on $\mathcal{I}$. The full analysis is done in exact, non-linear, general
relativity.
