A comprehensive angle-resolved photoemission spectroscopy study of the band structure in single-layer cuprates is presented with the aim of uncovering universal trends across different materials. Five different hole- and electron-overdoped cuprate superconductors (La1.59Eu0.2Sr0.21CuO4, La1.77Sr0.23CuO4, Bi1.74Pb0.38Sr1.88CuO6+δ, Tl2Ba2CuO6+δ, and Pr1.15La0.7Ce0.15CuO4) have been studied with special focus on the bands with a predominately d-orbital character. Using a light polarization analysis, the eg and t2g bands are identified across these materials. A clear correlation between the d3z2-r2 band energy and the apical oxygen distance dA is demonstrated. Moreover, the compound dependence of the dx2-y2 band bottom and the t2g band top is revealed. A direct comparison to density functional theory (DFT) calculations employing hybrid exchange-correlation functionals demonstrates excellent agreement. We thus conclude that the DFT methodology can be used to describe the global band structure of overdoped single-layer cuprates on both the hole- and electron-doped side.