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 dz2 and chalcogen pz 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.