  We present cosmological constraints from a joint analysis of the pre- and
post-reconstruction galaxy power spectrum multipoles from the final data
release of the Baryon Oscillation Spectroscopic Survey (BOSS). Geometric
constraints are obtained from the positions of BAO peaks in reconstructed
spectra, analyzed in combination with the unreconstructed spectra in a
full-shape (FS) likelihood using a joint covariance matrix, giving stronger
parameter constraints than FS-only or BAO-only analyses. We introduce a new
method for obtaining constraints from reconstructed spectra based on a
correlated theoretical error, which is shown to be simple, robust, and
applicable to any flavor of density-field reconstruction. Assuming $\Lambda$CDM
with massive neutrinos, we analyze data from two redshift bins
$z_\mathrm{eff}=0.38,0.61$ and obtain $1.6\%$ constraints on the Hubble
constant $H_0$, using only a single prior on the current baryon density
$\omega_b$ from Big Bang Nucleosynthesis (BBN) and no knowledge of the power
spectrum slope $n_s$. This gives $H_0 =
68.6\pm1.1\,\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$, with the inclusion of BAO
data sharpening the measurement by $40\%$, representing one of the strongest
current constraints on $H_0$ independent of cosmic microwave background data.
Restricting to the best-fit slope $n_s$ from Planck (but without additional
priors on the spectral shape), we obtain a $1\%$ $H_0$ measurement of $67.8\pm
0.7\,\mathrm{km\,s}^{-1}\mathrm{Mpc}^{-1}$. We find strong constraints on the
cosmological parameters from a joint analysis of the FS, BAO, and Planck data.
This sets new bounds on the sum of neutrino masses $\sum m_\nu <
0.14\,\mathrm{eV}$ (at $95\%$ confidence) and the effective number of
relativistic degrees of freedom $N_\mathrm{eff} = 2.90^{+0.15}_{-0.16}$, though
contours are not appreciably narrowed by the inclusion of BAO data.
