  We propose a solution to a classic problem in gravitational physics
consisting of defining the spin associated with asymptotically-flat spacetimes.
We advocate that the correct asymptotic symmetry algebra to approach this
problem is the generalized-BMS algebra $\textsf{gbms}$ instead of the BMS
algebra used hitherto in the literature for which a notion of spin is
generically unavailable. We approach the problem of defining the spin charges
from the perspective of coadjoint orbits of $\textsf{gbms}$ and construct the
complete set of Casimir invariants that determine $\textsf{gbms}$ coadjoint
orbits, using the notion of vorticity for $\textsf{gbms}$. This allows us to
introduce spin charges for $\textsf{gbms}$ as the generators of area-preserving
diffeomorphisms forming its isotropy subalgebra. To elucidate the parallelism
between our analysis and the Poincar\'e case, we clarify several features of
the Poincar\'e embedding in $\textsf{gbms}$ and reveal the presence of
condensate fields associated with the symmetry breaking from $\textsf{gbms}$ to
Poincar\'e. We also introduce the notion of a rest frame available only for
this extended algebra. This allows us to construct, from the spin generator,
the gravitational analog of the Pauli--Luba\'nski pseudo-vector. Finally, we
obtain the $\textsf{gbms}$ moment map, which we use to construct the
gravitational spin charges and gravitational Casimirs from their dual algebra
counterparts.
