Charged excitons, or X± trions, in monolayer transition-metal dichalcogenides have binding energies of several tens of meV. Together with the neutral exciton X0 they dominate the emission spectrum at low and elevated temperatures. We use charge-tunable devices based on WSe2 monolayers encapsulated in hexagonal boron nitride to investigate the difference in binding energy between X+ and X- and the X- fine structure. We find in the charge-neutral regime, the X0 emission accompanied at lower energy by a strong peak close to the longitudinal optical (LO) phonon energy. This peak is absent in reflectivity measurements, where only the X0 and an excited state of the X0 are visible. In the n-doped regime, we find a closer correspondence between emission and reflectivity as the trion transition with a well-resolved fine-structure splitting of 6 meV for X- is observed. We present a symmetry analysis of the different X+ and X- trion states and results of the binding energy calculations. We compare the trion binding energy for the n- and p-doped regimes with our model calculations for low carrier concentrations. We demonstrate that the splitting between the X+ and X- trions as well as the fine structure of the X- state can be related to the short-range Coulomb-exchange interaction between the charge carriers.