Transparent conducting oxides (TCO) are materials that combine electrical conductivity, with transparency around 90% in visible spectrum. Although these oxides have been deeply studied, several questions remain unsolved such as the role of the atomic radius and the stability of the gallia (Ga2O3) structure and an atomic-orbital understanding of the crystal structure, electronic states, symmetry, and chemical species in the disparity between the optical and fundamental band gap in In2O3 bixbyite. Those questions were addressed by first-principles calculations based on density functional theory, and a solid explanation will reported for the crystal stability of the gallia structure and three fundamental rules to explain the band gap disparity, which can be applied for every compound: (i) inversion symmetry in the crystal structure; (ii) conduction band minimum formed by cations and O
