Complexes formed by H and the isoelectronic impurity N in GaAs1-yNy alloys have been widely investigated because the significant effects of N on the GaAs properties and their passivation by H represent a unique tool for a defect engineering of semiconductors. However, available results still present a quite puzzling picture. Both the N-H2* and C2v complexes proposed by theory were challenged indeed by experimental results. In the present Letter, we disclose a double-faced behavior of a H atom interacting with an isoelectronic impurity: while H, on one side, binds to N and induces the formation of dangling bonds (DB) on its Ga neighbors, on the other side, it saturates these DBs, thus permitting the formation of multiple-H complexes. This peculiar H behavior fully explains the experimental findings and likely represents a general feature of H-isoelectronic-impurity interactions.