We predict linear polarization for a radioactively powered kilonova following the merger of a black hole and a neutron star. Specifically, we perform 3D Monte Carlo radiative transfer simulations for two different models, both featuring a lanthanide-rich dynamical ejecta component from numerical-relativity simulations while only one including an additional lanthanide-free disc-wind component. We calculate polarization spectra for nine different orientations at 1.5, 2.5, and 3.5 d after the merger and in the 0.1-2 mu m wavelength range. We find that both models are polarized at a detectable level 1.5 d after the merger while show negligible levels thereafter. The polarization spectra of the two models are significantly different. The model lacking a disc wind shows no polarization in the optical, while a signal increasing at longer wavelengths and reaching similar to 1-6 per cent at 2 mu m depending on the orientation. The model with a disc-wind component, instead, features a characteristic 'double-peak' polarization spectrum with one peak in the optical and the other in the infrared. Polarimetric observations of future events will shed light on the debated neutron richness of the disc-wind component. The detection of optical polarization would unambiguously reveal the presence of a lanthanide-free disc-wind component, while polarization increasing from zero in the optical to a peak in the infrared would suggest a lanthanide-rich composition for the whole ejecta. Future polarimetric campaigns should prioritize observations in the first similar to 48 h and in the 0.5-2 mu m range, where polarization is strongest, but also explore shorter wavelengths/later times where no signal is expected from the kilonova and the interstellar polarization can be safely estimated.