  The velocity anisotropy of particles inside dark matter (DM) haloes is an
important physical quantity, which is required for the accurate modelling of
mass profiles of galaxies and clusters of galaxies. It is typically measured
using the ratio of the radial-to-tangential velocity dispersions at a given
distance from the halo centre. However, this measure is insufficient to
describe the dynamics of realistic haloes, which are typically quite elongated.
Studying the velocity distribution in massive DM haloes in cosmological
simulations, we find that in the inner parts of the haloes the local velocity
ellipsoids are strongly aligned with the major axis of the halo, the alignment
being stronger for more relaxed haloes. In the outer regions of the haloes, the
alignment becomes gradually weaker and the orientation is more random. These
two distinct regions of different degree of the alignment coincide with two
characteristic regimes of the DM density profile: shallower and steeper than
\rho r^{-2}. This alignment of the local velocity ellipsoids requires
reinterpretation of features found in measurements based on the spherically
averaged ratio of the radial-to-tangential velocity dispersions. In particular,
we show that the velocity distribution in the central halo regions is highly
anisotropic. For cluster-size haloes with mass 10^{14}-10^{15} h^-1 Msun, the
velocity anisotropy along the major axis is nearly independent of radius and is
equal to \beta=1-\sigma^2_{perp}/\sigma^2_{radial}=0.4, which is significantly
larger than the previously estimated spherically averaged velocity anisotropy.
The alignment of density and velocity anisotropies, and the radial trends may
also have some implications for the mass modelling based on kinematical data of
such objects as galaxy clusters or dwarf spheroidals, where the orbital
anisotropy is a key element in an unbiased mass inference.
