  Charged excitons, or X$^{\pm}$-trions, in monolayer transition metal
dichalcogenides have binding energies of several tens of meV. Together with the
neutral exciton X$^0$ they dominate the emission spectrum at low and elevated
temperatures. We use charge tunable devices based on WSe$_2$ 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 X$^0$ 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 X$^0$ and an excited state
of the X$^0$ 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.
