  We present 353 GHz full-sky maps of the polarization fraction $p$, angle
$\psi$, and dispersion of angles $S$ of Galactic dust thermal emission produced
from the 2018 release of Planck data. We confirm that the mean and maximum of
$p$ decrease with increasing $N_H$. The uncertainty on the maximum polarization
fraction, $p_\mathrm{max}=22.0$% at 80 arcmin resolution, is dominated by the
uncertainty on the zero level in total intensity. The observed inverse
behaviour between $p$ and $S$ is interpreted with models of the polarized sky
that include effects from only the topology of the turbulent Galactic magnetic
field. Thus, the statistical properties of $p$, $\psi$, and $S$ mostly reflect
the structure of the magnetic field. Nevertheless, we search for potential
signatures of varying grain alignment and dust properties. First, we analyse
the product map $S \times p$, looking for residual trends. While $p$ decreases
by a factor of 3--4 between $N_H=10^{20}$ cm$^{-2}$ and $N_H=2\times 10^{22}$
cm$^{-2}$, $S \times p$ decreases by only about 25%, a systematic trend
observed in both the diffuse ISM and molecular clouds. Second, we find no
systematic trend of $S \times p$ with the dust temperature, even though in the
diffuse ISM lines of sight with high $p$ and low $S$ tend to have colder dust.
We also compare Planck data with starlight polarization in the visible at high
latitudes. The agreement in polarization angles is remarkable. Two polarization
emission-to-extinction ratios that characterize dust optical properties depend
only weakly on $N_H$ and converge towards the values previously determined for
translucent lines of sight. We determine an upper limit for the polarization
fraction in extinction of 13%, compatible with the $p_\mathrm{max}$ observed in
emission. These results provide strong constraints for models of Galactic dust
in diffuse gas.
