Mueller matrix spectroscopic ellipsometry is applied to determine anisotropic optical properties for a set of single-crystal rhombohedral structure α-(AlxGa1−x)2O3 thin films (0 ≤ x ≤ 1). Samples are grown by plasma-assisted molecular beam epitaxy on m-plane sapphire. A critical-point model is used to render a spectroscopic model dielectric function tensor and to determine direct electronic band-to-band transition parameters, including the direction dependent two lowest-photon energy band-to-band transitions associated with the anisotropic bandgap. We obtain the composition dependence of the direction dependent two lowest band-to-band transitions with separate bandgap bowing parameters associated with the perpendicular (bEg,⊥ = 1.31 eV) and parallel (bEg,|| = 1.61 eV) electric field polarization to the lattice c direction. Our density functional theory calculations indicate a transition from indirect to direct characteristics between α-Ga2O3 and α-Al2O3, respectively, and we identify a switch in band order where the lowest band-to-band transition occurs with polarization perpendicular to c in α-Ga2O3 whereas for α-Al2O3 the lowest transition occurs with polarization parallel to c. We estimate that the change in band order occurs at approximately 40% Al content. Additionally, the characteristic of the lowest energy critical point transition for polarization parallel to c changes from M1 type in α-Ga2O3 to M0 type van Hove singularity in α-Al2O3.