A Hamiltonian, H-d,H-(2), "rigorously" diabatic in the vicinity of R-x, a point of conical intersection, is constructed using second-order degenerate perturbation theory. Near an R-x on a C-2v seam of conical intersection of two states of different symmetry, H-d,H-(2) may exhibit a confluence with a C-s seam of conical intersection of two states of the same symmetry. Thus by construction, there exists a "rigorous" diabatic representation of the vicinity of this confluence. A procedure for defining a unique linear combination of the degenerate states at a conical intersection is found to be useful for determining the parameters for H-d,H-(2) and for identifying approximate symmetries in situations where point group symmetry is rigorously absent.