The fundamental molecular ion H3(+) has impacted astronomy, chemistry, and physics, particularly since the discovery of its rovibrational spectrum. Consisting of three identical fermions, its properties are profoundly influenced by the requirements of exchange symmetry, most notably the nonexistence of its ground rotational state. Spectroscopy of H3(+) is often used to infer the relative abundances of its two nuclear spin modifications, ortho- and para-H3(+), which are important in areas as diverse as electron dissociative recombination and deuterium fractionation in cold interstellar clouds. In this paper, we explore in detail the impact of exchange symmetry on the states of H3(+), with a particular focus on the state degeneracies necessary for converting spectral transition intensities to relative abundances. We address points of confusion in the literature surrounding these issues and discuss the implications for proton-transfer reactions of H3(+) at low temperatures.