Mueller matrix spectroscopic ellipsometry is applied to determine anisotropic optical properties for a set of single-crystal rhombohedral structure alpha-(AlxGa1-x)(2)O-3 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 (b(Eg,⊥) = 1.31 eV) and parallel (b(Eg,||) = 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 alpha-Ga2O3 and alpha-Al2O3, respectively, and we identify a switch in band order where the lowest band-to-band transition occurs with polarization perpendicular to c in alpha-Ga2O3 whereas for alpha-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 M-1 type in alpha-Ga2O3 to M-0 type van Hove singularity in alpha-Al2O3.