  We apply the Dirac procedure for constrained systems to the
Arnowitt-Deser-Misner formalism linearized around the Bianchi I universe. We
discuss and employ basic concepts such as Dirac observables, Dirac brackets,
gauge-fixing conditions, reduced phase space, physical Hamiltonian, canonical
isomorphism between different gauge-fixing surfaces and spacetime
reconstruction. We relate this approach to the gauge-fixing procedure for
non-perturbative canonical relativity. We discuss the issue of propagating a
basis for the scalar-vector-tensor decomposition as, in an anisotropic
universe, the wavefronts of plane waves undergo a nontrivial evolution. We show
that the definition of a gravitational wave as a traceless-transverse mode of
the metric perturbation needs to be revised. Moreover there exist coordinate
systems in which a polarization mode of the gravitational wave is given
entirely in terms of a scalar metric perturbation. We first develop the
formalism for the universe with a single scalar field and then extend it to the
multi-field case. The obtained fully canonical formalism will serve as a
starting point for a complete quantization of the cosmological perturbations
and the cosmological background.
