Ultracold bosons in optical lattices are one of the few systems where bosonic matter is known to exhibit strong correlations. Here we push the frontier of our understanding of interacting bosons in optical lattices by adding synthetic spin-orbit coupling, and show that new kinds of density and chiral orders develop. The competition between the optical lattice period and the spin-orbit coupling length-which can be made comparable in experiments-along with the spin hybridization induced by a transverse field (i.e., Rabi coupling) and interparticle interactions create a rich variety of quantum phases including uniform, nonuniform, and phase-separated superfluids, as well as Mott insulators. The spontaneous symmetry-breaking phenomena at the transitions between them are explained by a two-order-parameter Ginzburg-Landau model with multiparticle umklapp processes. Finally, in order to characterize each phase, we calculated their experimentally measurable crystal momentum distributions.