Many properties of layered materials change as they are thinned from their bulk forms down to single layers, with examples including indirect-to-direct band gap transition in 2H semiconducting transition metal dichalcogenides as well as thickness-dependent changes in the valence band structure in post-transition-metal monochalcogenides and black phosphorus. Here, we use angle-resolved photoemission spectroscopy to study the electronic band structure of monolayer ReSe₂, a semiconductor with a distorted 1T structure and in-plane anisotropy. By changing the polarization of incoming photons, we demonstrate that for ReSe₂, in contrast to the 2H materials, the out-of-plane transition metal d_z² and chalcogen p_z orbitals do not contribute significantly to the top of the valence band, which explains the reported weak changes in the electronic structure of this compound as a function of layer number. We estimate a band gap of 1.7 eV in pristine ReSe₂ using scanning tunneling spectroscopy and explore the implications on the gap following surface doping with potassium. A lower bound of 1.4 eV is estimated for the gap in the fully doped case, suggesting that doping-dependent many-body effects significantly affect the electronic properties of ReSe₂. Our results, supported by density functional theory calculations, provide insight into the mechanisms behind polarization-dependent optical properties of rhenium dichalcogenides and highlight their place among two-dimensional crystals.