  We present a statistical study of the relative orientation in the plane of
the sky between interstellar magnetic fields and filaments hosting cold clumps.
For the first time, we consider both the density of the environment and the
density contrast between the filaments and their environment. Moreover, we
geometrically distinguish between the clumps and the remaining portions of the
filaments. We infer the magnetic field orientations in the filaments and in
their environment from the Stokes parameters, assuming optically thin
conditions. Thus, we analyze the relative orientations between filaments,
embedded clumps, and internal and background magnetic fields, depending on the
filament environment and evolutionary stages. We recover the previously
observed trend for filaments in low column density environments to be aligned
parallel to the background magnetic field; however, we find that this trend is
significant only for low contrast filaments, whereas high contrast filaments
tend to be randomly orientated with respect to the background magnetic field.
Filaments in high column density environments do not globally show any
preferential orientation, although low contrast filaments alone tend to have
perpendicular relative orientation with respect to the background magnetic
field. For a subsample of nearby filaments, for which volume densities can be
derived, we find a clear transition in the relative orientation with increasing
density, at $n_{\rm H} \sim 10^{3}~{\rm cm}^{-3}$, changing from mostly
parallel to mostly perpendicular in the off-clump portions of filaments and
from even to bimodal in the clumps. Our results confirm a strong interplay
between interstellar magnetic fields and filaments during their formation and
evolution.
