  We derive a Hamiltonian formulation of the theory of gauge invariant, linear
perturbations in anisotropic Bianchi I spacetimes, and describe how to quantize
this system. The matter content is assumed to be a minimally coupled scalar
field with potential $V(\phi)$. We show that a Bianchi I spacetime generically
induces both anisotropies and quantum entanglement on cosmological
perturbations, and provide the tools to compute the details of these features.
We then apply this formalism to a scenario in which the inflationary era is
preceded by an anisotropic Bianchi I phase, and discuss the potential imprints
in observable quantities. The formalism developed here paves the road to a
simultaneous canonical quantization of both the homogeneous degrees of freedom
and the perturbations, a task that we develop in a companion paper.
