  Context. Turbulent transport in stellar radiative zones is a key ingredient
of stellar evolution theory, but the anisotropy of the transport due to the
stable stratification and the rotation of these regions is poorly understood.
The assumption of shellular rotation, which is a cornerstone of the so-called
rotational mixing, relies on an efficient horizontal transport. However, this
transport is included in many stellar evolution codes through phenomenological
models that have never been tested.
  Aims. We investigate the impact of horizontal shear on the anisotropy of
turbulent transport.
  Methods. We used a relaxation approximation (also known as {\tau}
approximation) to describe the anisotropising effect of stratification,
rotation, and shear on a background turbulent flow by computing velocity
correlations.
  Results. We obtain new theoretical scalings for velocity correlations that
include the effect of horizontal shear. These scalings show an enhancement of
turbulent motions, which would lead to a more efficient transport of chemicals
and angular momentum, in better agreement with helio- and asteroseismic
observations of rotation in the whole Hertzsprung-Russell diagram. Moreover, we
propose a new choice for the non-linear time used in the relaxation
approximation, which characterises the source of the turbulence.
  Conclusions. For the first time, we describe the effect of stratification,
rotation, and vertical and horizontal shear on the anisotropy of turbulent
transport in stellar radiative zones. The new prescriptions need to be
implemented in stellar evolution calculations. To do so, it may be necessary to
implement non-diffusive transport.
