Energy-band lineups at several [100] heterojunctions of III-V semiconductors are calculated using a self-consistent tight-binding treatment. The calculations exhibit transitivity to within 0.2 eV for ${\mathrm{In}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ga}}_{\mathit{x}}$As/${\mathrm{In}}_{1\mathrm{\ensuremath{-}}\mathit{y}}$${\mathrm{Al}}_{\mathit{y}}$As/InP, GaAs/InAs/InP, GaAs/GaP/InP, and GaSb/GaAs/InAs. For ${\mathrm{In}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Ga}}_{\mathit{x}}$As/${\mathrm{In}}_{1\mathrm{\ensuremath{-}}\mathit{y}}$${\mathrm{Al}}_{\mathit{y}}$As/InP, the results are in good agreement with experimental data. For heterojunctions where the constituents share neither a common anion nor cation, the two possible interfaces do not necessarily lead to a single-band offset. This, and also the strain configuration, has to be considered when applying the transitivity rule (that is, the fact that for three semiconductors, A, B, and C, the band offset at the heterojunction A/B can be deduced from the band offsets at the heterojunctions A/C and C/B), provided we take care that the material C corresponds to the material at the interface.