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We derive manifestly covariant actions of spinning particles starting fromcoadjoint orbits of isometry groups, by using Hamiltonian reductions. We showthat the defining conditions of a classical Lie group can be treated asHamiltonian 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 symplecticdual pair correspondence between the coadjoint orbits of the isometry group andthose of the dual Lie group, whose quantum version is the reductive dual paircorrespondence \`a la Howe. We show explicitly how various particle speciesarise from the classification of coadjoint orbits of Poincar\'e and (A)dSsymmetry. In the Poincar\'e case, we recover the data of the Wignerclassification, which includes continuous spin particles, (spinning) tachyonsand null particles with vanishing momenta, besides the usual massive andmassless spinning particles. In (A)dS case, our classification results are notonly consistent with the pattern of the corresponding unitary irreduciblerepresentations observed in the literature, but also contain novel information.In dS, we find the presence of partially massless spinning particles, butcontinuous spin particles, spinning tachyons and null particles are absent. TheAdS case shows the largest diversity of particle species. It has all particlesspecies of Poincar\'e symmetry except for the null particle, but allows inaddition various exotic entities such as one parameter extension of continuousparticles and conformal particles living on the boundary of AdS. Notably, wealso find a large class of particles living in "bitemporal" AdS space,including ones where mass and spin play an interchanged role. We also discussthe relative inclusion structure of the corresponding orbits.

Manifestly Covariant Worldline Actions from Coadjoint Orbits. Part I: Generalities and Vectorial Descriptions