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tokens: magnetic field and the brightness ridges on the sky. 2. Data sets Planck observed the sky polarization in seven frequency bands from 30 to 353 GHz ( Planck Collaboration I 2014 ). In this paper, we only use the data from the High Frequency Instrument (HFI, Lamarre et al. 2010 ) at the highest frequency, 353 GHz, where the dust emission is the brightest. This is the best-suited Planck map for studying the structure of the dust polarization sky ( Planck Collaboration Int. XIX 2015 ; Planck Collaboration Int. XX 2015 ). We use the Stokes and maps and the associated noise maps made with the five independent consecutive sky surveys of the Planck cryogenic mission, which correspond to the DR3 (delta-DX9) internal data release. We refer to previous Planck publications for the data processing, map-making, photometric calibration, and photometric uncertainties ( Planck Collaboration II 2014 ; Planck Collaboration VI 2014 ; Planck Collaboration V 2014 ; Planck Collaboration VIII 2014 ). The and maps are corrected for spectral leakage as described in Planck Collaboration Int. XIX (2015) . As in this first Planck polarization paper, we use the IAU convention for the polarization angle, measured from the local direction to the north Galactic pole with positive values towards the east. For the dust total intensity at 353 GHz we use the model map, , and the associated noise map, derived from a fit with a modified blackbody of the Planck data at GHz, and IRAS at m ( Planck Collaboration XI 2014 ). This map has a lower noise than the corresponding 353 GHz Stokes Planck map. Furthermore, is the dust specific intensity corrected for zodiacal emission, cosmic microwave background (CMB) anisotropies, and the cosmic infrared background (CMB) monopole. The and maps are initially at 48 resolution, and at . The three maps are in HEALPix format 2 with a pixelization . To increase the signal-to-noise ratio of extended emission, we smooth the three maps to resolution using the Gaussian approximation to the Planck beam. We reduce the HEALPix resolution to (71 pixels) after smoothing. To finely sample the beam, we also use the smoothed map with in Appendix A . For the polarization maps, we apply the “” routine of HEALPix that decomposes the and maps into and maps, applies the Gaussian smoothing in harmonic space, and transforms the smoothed and back into and maps at resolution. Most of our analysis is based on the , , and maps, but we also use the maps of the de-biased polarization fraction and angle produced by Planck Collaboration Int. XIX (2015) . The contribution of the CMB polarization to the and maps at 353 GHz is negligible for this study. 3. Structures in the polarization maps We introduce dust polarization (Sect. 3.1 ) and present all-sky images highlighting localized structures in the dust Stokes and maps correlated with corresponding features in intensity (Sect. 3.2 ). Fig. 1 Top : all-sky Mollweide display of the dust emission intensity at 353 GHz after