Abstract: We analyze the paradigmatic competition between intraband and crossband Cooper-pair formation in twoband superconductors, neglected in most works to date. We derive the phase-sensitive gap equations and describe the crossover between the intraband-dominated and the crossband-dominated regimes, delimited by a “gapless” state. Experimental signatures of crosspairing comprise notable gap splitting in the excitation spectrum, non-BCS behavior of gaps versus temperature, as well as changes in the pairing symmetry as a function of temperature. The consequences of these findings are illustrated on the examples of MgB2 and Ba0.6K0.4Fe2As2.