  We reveal an $\mathfrak{iso}(2,1)$ Poincar\'e algebra of conserved charges
associated with the dynamics of the interior of black holes. The action of
these Noether charges integrates to a symmetry of the gravitational system
under the Poincar\'e group ISO$(2,1)$, which allows to describe the evolution
of the geometry inside the black hole in terms of geodesics and horocycles of
AdS${}_2$. At the Lagrangian level, this symmetry corresponds to M\"obius
transformations of the proper time together with translations. Remarkably, this
is a physical symmetry changing the state of the system, which also naturally
forms a subgroup of the much larger
$\textrm{BMS}_{3}=\textrm{Diff}(S^1)\ltimes\textrm{Vect}(S^1)$ group, where
$S^1$ is the compactified time axis. It is intriguing to discover this
structure for the black hole interior, and this hints at a fundamental role of
BMS symmetry for black hole physics. The existence of this symmetry provides a
powerful criterion to discriminate between different regularization and
quantization schemes. Following loop quantum cosmology, we identify a
regularized set of variables and Hamiltonian for the black hole interior, which
allows to resolve the singularity in a black-to-white hole transition while
preserving the Poincar\'e symmetry on phase space. This unravels new aspects of
symmetry for black holes, and opens the way towards a rigorous group
quantization of the interior.
