  We derive manifestly covariant actions of spinning particles starting from
coadjoint orbits of isometry groups, by using Hamiltonian reductions. We show
that the defining conditions of a classical Lie group can be treated as
Hamiltonian constraints which generate the coadjoint orbits of another, dual,
Lie group. In case of (inhomogeneous) orthogonal groups, the dual groups are
(centrally-extended inhomogeneous) symplectic groups. This defines a symplectic
dual pair correspondence between the coadjoint orbits of the isometry group and
those of the dual Lie group, whose quantum version is the reductive dual pair
correspondence \`a la Howe. We show explicitly how various particle species
arise from the classification of coadjoint orbits of Poincar\'e and (A)dS
symmetry. In the Poincar\'e case, we recover the data of the Wigner
classification, which includes continuous spin particles, (spinning) tachyons
and null particles with vanishing momenta, besides the usual massive and
massless spinning particles. In (A)dS case, our classification results are not
only consistent with the pattern of the corresponding unitary irreducible
representations observed in the literature, but also contain novel information.
In dS, we find the presence of partially massless spinning particles, but
continuous spin particles, spinning tachyons and null particles are absent. The
AdS case shows the largest diversity of particle species. It has all particles
species of Poincar\'e symmetry except for the null particle, but allows in
addition various exotic entities such as one parameter extension of continuous
particles and conformal particles living on the boundary of AdS. Notably, we
also find a large class of particles living in "bitemporal" AdS space,
including ones where mass and spin play an interchanged role. We also discuss
the relative inclusion structure of the corresponding orbits.
