We present a global study of how the relative equilibria of the H-3(+) ion change as the angular momentum J increases. A relative equilibrium is a classical trajectory for which the molecule rotates about a stationary axis without changing its shape. The study confirms previous results which show that the geometry of the minimum energy relative equilibria changes from an equilateral triangle to a symmetric linear configuration at around J=47. The series of bifurcations and stability changes that accompany this transition is presented in detail. New results include the discovery that the rotating equilateral triangle remains linearly stable for a large range of angular momentum values beyond the point where it ceases to be a minimum of the total energy. A third type of relative equilibrium, a rotating isosceles triangle, is also found to be linearly stable in the approximate range J=0-34. Both the equilateral and isosceles triangle configurations lose stability via Hamiltonian-Hopf bifurcations. The frequencies and symmetry species of the normal modes of the stable relative equilibria are computed and harmonic quantization is used to predict how the symmetries of the lowest lying quantum states will change as J increases. Energy level clustering due to tunneling between symmetry-equivalent relative equilibria is described.