Knowing the band structure of materials is one of the prerequisites to understanding their properties. Therefore, angle-resolved photoemission spectroscopy (ARPES) has become a highly demanded experimental tool to investigate the band structure. However, especially in thin film materials with a layered structure and several capping layers, access to the electronic structure by ARPES is limited. Therefore, several alternative methods to obtain the required information have been suggested. Here we directly invert the results by cyclotron resonance experiments to obtain the band structure of a two-dimensional (2D) material. This procedure is applied to the mercury telluride quantum well with a critical thickness which is characterized by a 2D electron gas with linear dispersion relations. The Dirac-like band structure in this material could be mapped both on the electron and on the hole side of the band diagram. In this material, purely linear dispersion of the holelike carriers is in contrast to detectable quadratic corrections for the electrons.