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Polarized dust emission is a key tracer in the study of interstellar medium and of star formation. The observed polarization, however, is a product of magnetic field structure, dust grain properties, and grain alignment efficiency, as well as their variations in the line of sight, making it difficult to interpret polarization unambiguously. The comparison of polarimetry at multiple wavelengths is a possible way of mitigating this problem. We use data from HAWC+ /SOFIA and from SCUBA-2/POL-2 (from the BISTRO survey) to analyse the NGC 2071 molecular cloud at 154, 214, and 850 $\mu$m. The polarization angle changes significantly with wavelength over part of NGC 2071, suggesting a change in magnetic field morphology on the line of sight as each wavelength best traces different dust populations. Other possible explanations are the existence of more than one polarization mechanism in the cloud or scattering from very large grains. The observed change of polarization fraction with wavelength, and the 214-to-154 $\mu$m polarization ratio in particular, are difficult to reproduce with current dust models under the assumption of uniform alignment efficiency. We also show that the standard procedure of using monochromatic intensity as a proxy for column density may produce spurious results at HAWC+wavelengths. Using both long-wavelength (POL-2, 850 $\mu$m) and short-wavelength (HAWC+, $\lesssim 200\, \mu$m) polarimetry is key in obtaining these results. This study clearly shows the importance of multi-wavelength polarimetry at submillimetre bands to understand the dust properties of molecular clouds and the relationship between magnetic field and star formation.