We statistically evaluated the relative orientation between gas column density structures, inferred from Herschel submillimetre observations, and the magnetic field projected on the plane of sky, inferred from polarized thermal emission of Galactic dust observed by the Balloon-borne Large-Aperture Submillimetre Telescope for Polarimetry (BLASTPol) at 250, 350, and 500 μm, towards the Vela C molecular complex. First, we find very good agreement between the polarization orientations in the three wavelength-bands, suggesting that, at the considered common angular resolution of 3.´0 that corresponds to a physical scale of approximately 0.61 pc, the inferred magnetic field orientation is not significantly affected by temperature or dust grain alignment effects. Second, we find that the relative orientation between gas column density structures and the magnetic field changes progressively with increasing gas column density, from mostly parallel or having no preferred orientation at low column densities to mostly perpendicular at the highest column densities. This observation is in agreement with previous studies by the Planck collaboration towards more nearby molecular clouds. Finally, we find a correspondencebetween (a) the trends in relative orientation between the column density structures and the projected magnetic field; and (b) the shape of the column density probability distribution functions (PDFs). In the sub-regions of Vela C dominated by one clear filamentary structure, or "ridges", where the high-column density tails of the PDFs are flatter, we find a sharp transition from preferentially parallel or having no preferred relative orientation at low column densities to preferentially perpendicular at highest column densities. In the sub-regions of Vela C dominated by several filamentary structures with multiple orientations, or "nests", where the maximum values of the column density are smaller than in the ridge-like sub-regions and the high-column density tails of the PDFs are steeper, such a transition is also present, but it is clearly less sharp than in the ridge-like sub-regions. Both of these results suggest that the magnetic field is dynamically important for the formation of density structures in this region.