The surfaces of the atmosphere-less objects of our solar system are traditionally probed via reflectance measurements and/or broadband linear polarimetry. Little attention has been paid so far to the wavelength dependence of the linear polarization of the scattered light. We decided to explore the potential of spectropolarimetry as a remote sensing tool for asteroids in addition to the more traditional reflectance measurements, and we carried out a spectropolarimetric survey of asteroids – to our best knolwedge, the first of its kind. We observed a sample of asteroids of different albedo and taxonomic classes, as well as a few regions at the limb of the Moon. We show that objects exhibiting similar reflectance spectra may display totally different polarization spectra, and we suggest that both intensity and polarization spectra should be used for asteroid classification. We also found that in some cases the Umov law is violated, that is, in contrast to what is expected from simple physical considerations, the fraction of linear polarization and the reflectance spectra may be correlated positively. We conclude that future modelling attempts of the surface structure of asteroids should be aimed at explaining both reflectance and polarization spectra.