We predict novel interband physics for bosons in double-well optical lattices (OLs). An intrinsic coupling between the s and the p(x) bands due to interaction gives rise to larger Mott regions on the phase diagram at even fillings than the ones at odd fillings. On the other hand, the ground state can form various types of condensates, including a mixture of single-particle condensates of both bands, a mixture of a single-particle condensate of one band and a pair condensate of the other band, and a pair condensate composed of one particle from one band and one hole from the other band. The predicted phenomena should be observable in current experiments on double-well OLs.