When the spin-orbit interaction is included, the character of a conical intersection in a molecule with an odd number of electrons differs dramatically from that of its nonrelativistic counterpart. In contrast to the two-dimensional branching space (eta=2) in the nonrelativistic case, for these conical intersections the branching space is five-dimensional (eta=5) in general, or three-dimensional (eta=3) when C-s symmetry is present. Recently we have introduced an algorithm, based on analytic gradient techniques, to locate such conical intersections and used related techniques to efficiently construct and study the properties of the vectors defining the branching space. Here we extend this analysis. A perturbative description of the eta=3 case is reported and used to determine the energy, derivative couplings, and a "rigorous" diabatic basis in the vicinity of a conical intersection. The perturbative results are compared with those of exact numerical calculations employing model Hamiltonians. The implications for the nuclear motion problem are discussed.