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
right asymptotic symmetry algebra to approach this problem is the generalized–BMS algebra
gbms, instead of the BMS algebra used hitherto in the literature. We approach the problem
of defining the spin charge from the perspective of coadjoint orbits of gbms. We construct
the complete set of Casimir invariants that determine gbms coadjoint orbits, using the notion
of vorticity for gbms. This allows us to introduce the spin charge for gbms as the generator
of its isotropy subalgebra. In order to elucidate the parallelism between our analysis and the
Poincar ́e case, we clarify several features of the Poincaré embedding in gbms and introduce the
notion of rest frame available only for this extended algebra. This allows us to construct, from
the spin generator, the gravitational analog of the Pauli–Lubánski pseudo-vector.