Lead halide perovskite nanocrystals are promising materials for classical and quantum light emission. To understand these outstanding properties, a thorough analysis of the band-edge exciton emission is needed which is not reachable in ensemble and room temperature studies because of broadening effects. Here, we report on a cryogenic-temperature study of the photoluminescence of single CsPbBr$_3$ NCs in the intermediate quantum confinement regime. We reveal the size-dependence of the spectral features observed: the bright-triplet exciton energy splittings, the trion and biexciton binding energies as well as the optical phonon replica spectrum. In addition, we show that bright triplet energy splittings are consistent with a pure exchange model and that the variety of polarisation properties and spectra recorded can be rationalised simply by considering the orientation of the emitting dipoles and the populations of the emitting states.
