  This article describes the theory of cosmological perturbations around a
homogeneous and anisotropic universe of the Bianchi I type. Starting from a
general parameterisation of the perturbed spacetime a la Bardeen, a complete
set of gauge invariant variables is constructed. Three physical degrees of
freedom are identified and it is shown that, in the case where matter is
described by a scalar field, they generalize the Mukhanov-Sasaki variables. In
order to show that they are canonical variables, the action for the
cosmological perturbations at second order is derived. Two major physical
imprints of the primordial anisotropy are identified: (1) a scalar-tensor
``see-saw'' mechanism arising from the fact that scalar, vector and tensor
modes do not decouple and (2) an explicit dependence of the statistical
properties of the density perturbations and gravity waves on the wave-vector
instead of its norm. This analysis extends, but also sheds some light on, the
quantization procedure that was developed under the assumption of a
Friedmann-Lemaitre background spacetime, and allows to investigate the
robustness of the predictions of the standard inflationary scenario with
respect to the hypothesis on the symmetries of the background spacetime. These
effects of a primordial anisotropy may be related to some anomalies of the
cosmic microwave background anisotropies on large angular scales.
