  We classify radial timelike geodesic motion of the exterior non-extremal Kerr
spacetime by performing a taxonomy of inequivalent root structures of the first
order radial geodesic equation using a novel compact notation and by
implementing the constraints from polar, time and azimuthal motion. Four
generic root structures with only simple roots give rise to eight non-generic
root structures when either one root becomes coincident with the horizon, one
root vanishes or two roots becomes coincident. We derive the explicit phase
space of all such root systems in the basis of energy, angular momentum and
Carter's constant and classify whether each corresponding radial geodesic
motion is allowed or disallowed from existence of polar, time and azimuthal
motion. The classification of radial motion within the ergoregion for both
positive and negative energies leads to 6 distinguished values of the Kerr
angular momentum. The classification of null radial motion and near-horizon
extremal Kerr radial motion are obtained as limiting cases and compared with
the literature. We explicitly parametrize the separatrix describing root
systems with double roots as the union of the following three regions that are
described by the same quartic respectively obtained when (1) the pericenter of
bound motion becomes a double root; (2) the eccentricity of bound motion
becomes zero; (3) the turning point of unbound motion becomes a double root.
