  It was found recently that the anisotropies in the homogeneous Bianchi I
cosmology considered within the context of a specific Horndeski theory are
damped near the initial singularity instead of being amplified. In this work we
extend the analysis of this phenomenon to cover the whole of the Horndeski
family. We find that the phenomenon is absent in the K-essence and/or Kinetic
Gravity Braiding theories, where the anisotropies grow as one approaches the
singularity. The anisotropies are damped at early times only in more general
Horndeski models whose Lagrangian includes terms quadratic and cubic in second
derivatives of the scalar field. Such theories are often considered as being
inconsistent with the observations because they predict a non-constant speed of
gravitational waves. However, the predicted value of the speed at present can
be close to the speed of light with any required precision, hence the theories
actually agree with the present time observations. We consider two different
examples of such theories, both characterized by a late self-acceleration and
an early inflation driven by the non-minimal coupling. Their anisotropies show
a maximum at intermediate times and approach zero at early and late times. The
early inflationary stage exhibits an instability with respect to inhomogeneous
perturbations, suggesting that the initial state of the universe should be
inhomogeneous. However, more general Horndeski models may probably be stable.
