  We present a general formalism for the Hamiltonian description of
perturbation theory around any spatially homogeneous spacetime. We employ and
refine the Dirac method for constrained systems, which is very well-suited to
cosmological perturbations. This approach includes a discussion of the
gauge-invariant dynamics of perturbations as well as an analysis of gauge
transformations, gauge-fixing, partial gauge-fixing and spacetime
reconstruction. We will introduce the Kucha\v{r} parametrization of the
kinematical phase space as a convenient tool for studying the gauge
transformations. The key element of this approach is the reconstruction of
spacetime based on gauge-fixing conditions.
