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tokens: scales, might be interpreted in terms of a decrease of with increasing column density ( Matthews et al. 2001 ; Whittet et al. 2008 ). A leading theory for the process of dust grain alignment involves radiative torques by the incident radiation ( Lazarian Hoang 2007 ; Hoang Lazarian 2009 ; Andersson 2015 ). A critical parameter for this mechanism is the ratio between the dust grain size and the radiation wavelength. As the dust column density increases, only the longer wavelength radiation penetrates the cloud and the alignment decreases. Grains within a cloud (without embedded sources) should have lower than those at the periphery of the same cloud. There is evidence for this from near-infrared interstellar polarization and submillimetre polarization along lines of sight through starless cores ( Jones et al. 2015 ), albeit on smaller scales and higher column densities than considered here. If inside the cloud is very low, the observed polarized intensity would arise from the dust in the outer layers, tracing the magnetic field in the “skin” of the cloud. Then the observed orientation of is not necessarily correlated with the column density structure, which is seen in total intensity, or with the magnetic field deep in the cloud. Soler et al. (2013) presented the results of HRO analysis on a series of synthetic observations produced using models of how might decrease with increasing density. They showed that with a steep decrease there is no visible correlation between the inferred magnetic field orientation and the high- structure, corresponding to nearly flat HROs. The HRO analysis of MCs carried out here reveals a correlation between the polarization orientation and the column density structure. This suggests that the dust polarized emission samples the magnetic field structure homogeneously on the scales being probed at the resolution of the Planck observations or, alternatively, that the field deep within high- structures has the same orientation of the field in the skin. 6. Conclusions We have presented a study of the relative orientation of the magnetic field projected on the plane of the sky (), as inferred from the Planck dust polarized thermal emission, with respect to structures detected in gas column density (). The relative orientation study was performed by using the histogram of relative orientations (HRO), a novel statistical tool for characterizing extended polarization maps. With the unprecedented statistics of polarization observations in extended maps obtained by Planck , we analyze the HRO in regions with different column densities within ten nearby molecular clouds (MCs) and two test fields. In most of the regions analysed we find that the relative orientation between and structures changes systematically with from being parallel in the lowest column density areas to perpendicular in the highest column density areas. The switch occurs at . This change in relative orientation is particularly significant given that projection