First-principles full-potential linearized augmented plane wave calculations have been performed for lattice mismatched common-atom LII-V interfaces. In particular, we have examined the effects of epitaxial strain and ordering direction on the valence-band offset in [001] and [111] GaSb/InSb and InAs/InSb superlattices, and found that the valence-band maximum is always higher at the InSb side of the heterojunction, except for the common-anion system grown on an InSb substrate. The comparison between equivalent structures having the same substrate lattice constant, but different growth axis, shows that for comparable strain conditions, the ordering direction slightly influences the band lineup, due to small differences of the charge readjustment at the [001] and [111] interfaces. On the other hand, strain is shown to strongly affect the valence-band offset; in particular, as the pseudomorphic growth conditions are varied, the bulk contribution to the band lineup changes markedly, whereas the interface term is almost constant. On the whole, our calculations yield a band lineup that decreases linearly as the substrate lattice constant is increased, showing its high tunability as a function of different pseudomorphic growth conditions. Finally, the band lineup at the lattice matched InAs/GaSb interface determined using the transitivity rule gave perfect agreement between predicted and experimental results.