To investigate the presence of the Laing-Garrington effect (i.e. the l
obe containing the jet more depolarized than that containing the count
er-jet), we analyse polarization data at 6 and 20 cm for a sample of d
ouble low-luminosity radio galaxies. The effect is well visible, altho
ugh not very strong. We therefore confirm what was inferred in a previ
ous paper based on data at one frequency (20 cm) only. Furthermore, we
find that radio galaxies with strong radio cores show a more pronounc
ed asymmetry in depolarization. We conclude that the presence of the L
aing-Garrington effect in our sample is a indication that relativistic
motions do occur on the kpc scale of low-luminosity radio galaxies as
recently suggested by theoretical arguments and observational evidenc
e.