Revealing the crystal structure of lead halide perovskite nanocrystals is essential for the optimization of stability of these emerging materials in applications such as solar cells, photodetectors, and light-emitting devices. We use magneto-photoluminescence spectroscopy of individual perovskite CsPbBr3 nanocrystals as a unique tool to determine their crystal structure, which imprints distinct signatures in the excitonic sublevels of charge complexes at low temperatures. At zero magnetic field, the identification of two classes of photoluminescence spectra, displaying either two or three sublevels in their exciton fine structure, shows evidence for the existence of two crystalline structures, namely tetragonal D4h and orthorhombic D2h phases. Magnetic field shifts, splitting, and coupling of the sublevels provide a determination of the diamagnetic coefficient and valuable information on the exciton g-factor and its anisotropic character. Moreover, this spectroscopic study reveals the optical properties of charged excitons and allows the extraction of the electron and hole g-factors for perovskite systems.