  We use $(121\pm1)$ million $\Upsilon(3S)$ and $(98\pm1)$ 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
$\chi_{b\mathrm{J}}(1P,2P)$ 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
$\chi_{b0}(2P)\rightarrow\gamma\Upsilon(2S)$ and
$\chi_{b0}(1P)\rightarrow\gamma\Upsilon(1S)$ 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 $\Upsilon(3S)\rightarrow\gamma\chi_{b1,2}(1P)$ 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.
