We use (121±1) million Υ(3S) and (98±1) million Υ(2S) mesons recorded by the BABAR detector at the PEP-II e+e- collider at SLAC to perform a study of radiative transitions involving the χbJ(1P,2P) states in exclusive decays with μ+μ-γγ final states. We reconstruct twelve channels in four cascades using two complementary methods. In the first we identify both signal photon candidates in the electromagnetic calorimeter (EMC), employ a calorimeter timing-based technique to reduce backgrounds, and determine branching-ratio products and fine mass splittings. These results include the best observational significance yet for the χb0(2P)→γΥ(2S) and χb0(1P)→γΥ(1S) transitions. In the second method, we identify one photon candidate in the EMC and one which has converted into an e+e- pair due to interaction with detector material, and we measure absolute product branching fractions. This method is particularly useful for measuring Υ(3S)→γχb1,2(1P) decays. Additionally, we provide the most up-to-date derived branching fractions, matrix elements and mass splittings for χb transitions in the bottomonium system. Using a new technique, we also measure the two lowest-order spin-dependent coefficients in the nonrelativistic QCD Hamiltonian.