Molybdenum ditelluride, MoTe₂, is a versatile material where the topological phase can be readily tuned by manipulating the associated structural phase transition. The fine details of the band structure of MoTe₂, key to understanding its topological properties, have proven difficult to disentangle experientially due to the multiband character of the material. Through experimental optical conductivity spectra, we detect two strong low-energy interband transitions. Both are linked to excitations between spin-orbit split bands. The lowest interband transition shows a strong thermal shift, pointing to a chemical potential that dramatically decreases with temperature. With the help of ab initio calculations and a simple two-band model, we give qualitative and quantitative explanations of the main features in the temperature-dependent optical spectra up to 400 meV.