Observations of the submillimetre emission from Galactic dust, in both total intensityIand polarization, have received tremendous interest thanks to thePlanckfull-sky maps. In this paper we make use of such full-sky maps of dust polarized emission produced from the third public release ofPlanckdata. As the basis for expanding on astrophysical studies of the polarized thermal emission from Galactic dust, we present full-sky maps of the dust polarization fractionp, polarization angleψ, and dispersion function of polarization angles 𝒮. The joint distribution (one-point statistics) ofpandNHconfirms that the mean and maximum polarization fractions decrease with increasingNH. The uncertainty on the maximum observed polarization fraction,pmax= 22.0−1.4+3.5% at 353 GHz and 80′ resolution, is dominated by the uncertainty on the Galactic emission zero level in total intensity, in particular towards diffuse lines of sight at high Galactic latitudes. Furthermore, the inverse behaviour betweenpand 𝒮 found earlier is seen to be present at high latitudes. This follows the 𝒮 ∝ p−1relationship expected from models of the polarized sky (including numerical simulations of magnetohydrodynamical turbulence) that include effects from only the topology of the turbulent magnetic field, but otherwise have uniform alignment and dust properties. Thus, the statistical properties ofp,ψ, and 𝒮 for the most part reflect the structure of the Galactic magnetic field. Nevertheless, we search for potential signatures of varying grain alignment and dust properties. First, we analyse the product map 𝒮 × p, looking for residual trends. While the polarization fractionpdecreases by a factor of 3−4 betweenNH = 1020 cm−2andNH = 2 × 1022 cm−2, out of the Galactic plane, this product 𝒮 × ponly decreases by about 25%. Because 𝒮 is independent of the grain alignment efficiency, this demonstrates that the systematic decrease inpwithNHis determined mostly by the magnetic-field structure and not by a drop in grain alignment. This systematic trend is observed both in the diffuse interstellar medium (ISM) and in molecular clouds of the Gould Belt. Second, we look for a dependence of polarization properties on the dust temperature, as we would expect from the radiative alignment torque (RAT) theory. We find no systematic trend of 𝒮 × pwith the dust temperatureTd, whether in the diffuse ISM or in the molecular clouds of the Gould Belt. In the diffuse ISM, lines of sight with high polarization fractionpand low polarization angle dispersion 𝒮 tend, on the contrary, to have colder dust than lines of sight with lowpand high 𝒮. We also compare thePlanckthermal dust polarization with starlight polarization data in the visible at high Galactic latitudes. The agreement in polarization angles is remarkable, and is consistent with what we expect from the noise and the observed dispersion of polarization angles in the visible on the scale of thePlanckbeam. The two polarization emission-to-extinction ratios,RP/pandRS/V, which primarily characterize dust optical properties, have only a weak dependence on the column density, and converge towards the values previously determined for translucent lines of sight. We also determine an upper limit for the polarization fraction in extinction,pV/E(B − V), of 13% at high Galactic latitude, compatible with the polarization fractionp ≈ 20% observed at 353 GHz. Taken together, these results provide strong constraints for models of Galactic dust in diffuse gas.