Context. The analysis of the Planck polarization E and B mode power spectra of interstellar dust emission at 353 GHz recently raised new questions concerning the impact of Galactic foregrounds on the detection of the polarization of the cosmic microwave background (CMB) and on the physical properties of the interstellar medium (ISM). In the diffuse ISM at high latitude a clear E–B asymmetry is observed that has twice as much power in E modes as in B modes; there is also a positive correlation between the total power, T, and both E and B modes, which is currently interpreted in terms of the link between the structure of interstellar matter and that of the Galactic magnetic field.Aims. In this paper we aim to extend the Planck analysis of the high latitude sky to low Galactic latitudes, investigating the correlation between the T–E–B auto- and cross-correlation power spectra with the gas column density from the diffuse ISM to molecular clouds.Methods. We divided the sky between Galactic latitudes |b| > 5° and |b| < 60° in 552 circular patches, with an area of ~400°2, and we studied the cross-correlations between the T–E–B power spectra and the column density of each patch using the latest release of the Planck polarization data.Results. We find that the B-to-E power ratio (DlBB/DlEE) and the TE correlation ratio (rTE) depend on column density. While the former increases going from the diffuse ISM to molecular clouds in the Gould Belt, the latter decreases. This systematic variation must be related to actual changes in ISM properties. The data show significant scatter about this mean trend. The variations of DlBB/DlEE and rTE are observed to be anticorrelated for all column densities. In the diffuse ISM, the variance of these two ratios is consistent with a stochastic non-Gaussian model in which the values of DlBB/DlEE and rTE are fixed. We finally discuss the dependences of TB and EB with column density, which are however hampered by instrumental noise.Conclusions. For the first time, this work shows significant variations of the T–E–B power spectra of dust polarized emission across a large portion of the Galaxy. Their dependence on multipole and gas column density is key for accurate forecasts of next generation CMB experiments and for constraining present models of ISM physics (i.e., dust properties and interstellar turbulence), which are considered responsible for the observed T–E–B signals.Key words: dust, extinction / polarization / ISM: magnetic fields / cosmic background radiation / local insterstellar matter / ISM: clouds★ Based on observations obtained with Planck (http://www.esa.int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada.