  The phase space of general relativity in a finite subregion is characterized
by edge modes localized at the codimension-2 boundary, transforming under an
infinite-dimensional group of symmetries. The quantization of this symmetry
algebra is conjectured to be an important aspect of quantum gravity. As a step
towards quantization, we derive a complete classification of the positive-area
coadjoint orbits of this group for boundaries that are topologically a
2-sphere. This classification parallels Wigner's famous classification of
representations of the Poincar\'e group since both groups have the structure of
a semidirect product. We find that the total area is a Casimir of the algebra,
analogous to mass in the Poincar\'e group. A further infinite family of
Casimirs can be constructed from the curvature of the normal bundle of the
boundary surface. These arise as invariants of the little group, which is the
group of area-preserving diffeomorphisms, and are the analogues of spin.
Additionally, we show that the symmetry group of hydrodynamics appears as a
reduction of the corner symmetries of general relativity. Coadjoint orbits of
both groups are classified by the same set of invariants, and, in the case of
the hydrodynamical group, the invariants are interpreted as the generalized
enstrophies of the fluid.
