  (Abridged). It has been recently shown that random motions of the neutral
Hydrogen gas of the Triangulum galaxy (M33) exhibit a bisymmetric perturbation
which is aligned with the minor axis of the galaxy, suggesting a projection
effect. To investigate if perturbations in the velocity dispersion of nearby
discs are comparable to those of M33, the sample is extended to 32 galaxies
from The HI Nearby Galaxy Survey and the Westerbork HI Survey of Spiral and
Irregular Galaxies. We study velocity asymmetries in the disc planes by
performing Fourier transforms of high-resolution HI velocity dispersion maps
corrected for beam smearing effects, and measure the amplitudes and phase
angles of the Fourier harmonics. We find strong perturbations of first, second
and fourth orders. The strongest asymmetry is the bisymmetry, which is
predominantly associated with the presence of spiral arms. The first order
asymmetry is generally oriented close to the disc major axis, and the second
and fourth order asymmetries are preferentially oriented along intermediate
directions between the major and minor axes of the discs. These results are
evidence that strong projection effects shape the HI velocity dispersion maps.
The most likely source of systematic orientations is the anisotropy of
velocities, through the projection of streaming motions stronger along one of
the planar directions in the discs. Moreover, systematic phase angles of
asymmetries in the HI velocity dispersion could arise from tilted velocity
ellipsoids. We expect a larger incidence of correlation between the radial and
tangential velocities of HI gas. Our methodology is a powerful tool to
constrain the dominant direction of streaming motions and thus the shape of the
velocity ellipsoid of HI gas, which is de facto anisotropic at the angular
scales probed by the observations.
