Conduction and valence band offsets are amongst the most crucial material parameters for semiconductor heterostructure device design, such as for high-electron mobility transistors or quantum-well infrared photodetectors. Due to its expected high spontaneous electrical polarization and the possibility of polarization doping at heterointerfaces similar to the AlGaN/InGaN/GaN system, the metastable orthorhombic κ-phase of Ga2O3 and its indium and aluminum alloy systems are a promising alternative for such device applications. However, respective band offsets to any dielectric are unknown, as well as the evolution of the bands within the alloy systems. We report on valence and conduction band offsets of orthorhombic κ-(AlxGa1-x)2O3 and κ-(InxGa1-x)2O3 thin films to MgO as reference dielectric by X-ray photoelectron spectroscopy. The thin films with compositions xIn ≤ 0.27 and xAl ≤ 0.55 were grown by pulsed laser deposition utilizing tin-doped and radially-segmented targets. The determined band alignments reveal the formation of a type I heterojunction to MgO for all compositions with conduction band offsets of at least 1.4 eV, providing excellent electron confinement. Only low valence band offsets with a maximum of ~300 meV were observed. Nevertheless, this renders MgO as promising gate dielectric for metaloxide-semiconductor transistors in the orthorhombic modification. We further found that the conduction band offsets in the alloy systems are mainly determined by the evolution of the bandgaps, which can be tuned by the composition in a wide range between 4.1 and 6.2 eV, since the energy position of the valence band maximum stays almost constant over the complete composition range investigated. Therefore, tunable conduction band offsets of up to 1.1 eV within the alloy systems allow for subniveau transition energies in (AlxGa1-x)2O3/(InxGa1-x)2O3/(AlxGa1x)2O3 quantum wells from the IR to the visible regime, which are promising for application in quantum-well infrared photodetectors.