We find all free, massless, finite-dimensional, irreducible representations of the conformal group SO(D, 2) in all space-time dimensions D. In odd D they are only the scalar and spinor, but in even D they are all those which are (anti-)self-dual (chiral) on all vector indices. These are exactly the ones described by the mechanics of the extended spinning particle.
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Warren Siegel (1989) studied this question.