Highly reddened type Ia supernovae (SNe Ia) with low total-to-selective visual extinction ratio values, R
V, also show peculiar linear polarization wavelength dependencies with peak polarizations at short wavelengths (λ
max ≲ 0.4 μm). It is not clear why sightlines to SNe Ia display such different continuum polarization profiles from interstellar sightlines in the Milky Way with similar R
V values. We investigate polarization profiles of a sample of Galactic stars with low R
V values, along anomalous extinction sightlines, with the aim to find similarities to the polarization profiles that we observe in SN Ia sightlines. We undertook spectropolarimetry of 14 stars, used archival data for 3 additional stars, and ran dust extinction and polarization simulations (by adopting the picket-fence alignment model) to infer a simple dust model (size distribution, alignment) that can reproduce the observed extinction and polarization curves. Our sample of Galactic stars with low R
V values and anomalous extinction sightlines displays normal polarization profiles with an average λ
max 0.53 μm, and is consistent within 3σ to a larger coherent sample of Galactic stars from the literature. Despite the low R
V values of dust toward the stars in our sample, the polarization curves do not show any similarity to the continuum polarization curves observed toward SNe Ia with low R
V values. There is a correlation between the best-fit Serkowski parameters K and λ
max, but we did not find any significant correlation between R
V and λ
max. Our simulations show that the K-λ
max relationship is an intrinsic property of polarization. Furthermore, we have shown that in order to reproduce polarization curves with normal λ
max and low R
V values, a population of large (a ≥ 0.1μm) interstellar silicate grains must be contained in the dust composition. The reduced spectra are only available at the CDS via anonymous ftp to <A href="http://cdsarc.u-strasbg.fr">http://cdsarc.u-strasbg.fr</A> (ftp://130.79.128.5) or via <A href="http://cdsarc.u-strasbg.fr/viz- bin/qcat?J/A+A/615/A42">http://cdsarc.u-strasbg.fr/viz- bin/qcat?J/A+A/615/A42</A>
