  We examine the effects of spatial inhomogeneities on irrotational anisotropic
cosmologies by looking at the average properties of anisotropic pressure-free
models. Adopting the Buchert scheme, we recast the averaged scalar equations in
Bianchi-type form and close the standard system by introducing a propagation
formula for the average shear magnitude. We then investigate the evolution of
anisotropic average vacuum models and those filled with pressureless matter. In
the latter case we show that the backreaction effects can modify the familiar
Kasner-like singularity and potentially remove Mixmaster-type oscillations. The
presence of nonzero average shear in our equations also allows us to examine
the constraints that a phase of backreaction-driven accelerated expansion might
put on the anisotropy of the averaged domain. We close by assessing the status
of these and other attempts to define and calculate `average' spacetime
behaviour in general relativity.
