We investigate the group structure of the intrinsic dynamical variables describing the relativistic motions of the classical pure gyroscope. It is shown that the 10 elements of this group, the 6 components of the spin angular momentum tensor, and the four-velocity components have Poisson bracket relations among themselves characteristic of the Lie algebra of the De Sitter group. This algebraic result allows a complete description of the free particle motions to be deduced from a proper-time Hamiltonian linear in the four-momentum components. Thus we are led, via the correspondence principle, to a classical understanding of the origin of the algebraic and dynamical properties characteristic of Dirac-like relativistic wave equations.