We use (121 +/- 1) million Upsilon(3S) and (98 +/- 1) million Upsilon(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 chi(bJ)(1P, 2P) states in exclusive decays with mu(+)mu(-)gamma gamma 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 chi(b0)(2P) -> gamma Upsilon(2S) and chi(b0)(1P) -> gamma Upsilon(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 Upsilon(3S) -> gamma chi(b1,2)(1P) decays. Additionally, we provide the most up-to-date derived branching fractions, matrix elements and mass splittings for chi(b) transitions in the bottomonium system. Using a new technique, we also measure the two lowest-order spin-dependent coefficients in the nonrelativistic QCD Hamiltonian.