The Planck survey provides unprecedented full-sky coverage of the submillimetre polarized emission from Galactic dust. In addition to the information on the direction of the Galactic magnetic field, this also brings new constraints on the properties of dust. The dust grains that emit the radiation seen by Planck in the submillimetre also extinguish and polarize starlight in the visible. Comparison of the polarization of the emission and of the interstellar polarization on selected lines of sight probed by stars provides unique new diagnostics of the emission and light scattering properties of dust, and therefore of the important dust model parameters, composition, size, and shape. Using ancillary catalogues of interstellar polarization and extinction of starlight, we obtain the degree of polarization, pV, and the optical depth in the V band to the star, τV. Toward these stars we measure the submillimetre polarized intensity, PS, and total intensity, IS, in the Planck 353 GHz channel. We compare the column density measure in the visible, E(B - V), with that inferred from the Planck product map of the submillimetre dust optical depth and compare the polarization direction (position angle) in the visible with that in the submillimetre. For those lines of sight through the diffuse interstellar medium with comparable values of the estimated column density and polarization directions close to orthogonal, we correlate properties in the submillimetre and visible to find two ratios, RS /V = (PS/IS) / (pV/τV) and RP/p = PS/pV, the latter focusing directly on the polarization properties of the aligned grain population alone.We find RS /V = 4.2, with statistical and systematic uncertainties 0.2 and 0.3, respectively, and RP/p = 5.4 MJy sr-1, with uncertainties 0.2 and 0.3 MJy sr-1, respectively. Our estimate of RS /V is compatible with predictions based on a range of polarizing dust models that have been developed for the diffuse interstellar medium. This estimate provides new empirical validation of many of the common underlying assumptions of the models, but is not yet very discriminating among them. However, our estimate of RP/p is not compatible with predictions, which are too low by a factor of about 2.5. This more discriminating diagnostic, RP/p, indicates that changes to the optical properties in the models of the aligned grain population are required. These new diagnostics, together with the spectral dependence in the submillimetre from Planck,will be important for constraining and understanding the full complexity of the grain models, and for interpreting the Planck thermal dust polarization and refinement of the separation of this contamination of the cosmic microwave background. Appendices are available in electronic form at http://www.aanda.org