When are complements and substitutes dual?

For binary-action games on bipartite networks, strategic complements and strategic substitutes are one and the same: each can be transformed into the other by relabelling the actions of every player on exactly one side. In this note, I show that bipartiteness is necessary and sufficient for such a relabelling; as a consequence, bipartite networks are precisely those on which every binary-action game with strategic substitutes must have a pure-strategy Nash equilibrium.