  In this paper we examine in detail the implementation, with its associated
difficulties, of the Killing conditions and gauge fixing into the variational
principle formulation of Bianchi-Type cosmologies. We address problems raised
in the literature concerning the Lagrangian and the Hamiltonian formulations:
We prove their equivalence, make clear the role of the Homogeneity Preserving
Diffeomorphisms in the phase space approach, and show that the number of
physical degrees of freedom is the same in the Hamiltonian and Lagrangian
formulations. Residual gauge transformations play an important role in our
approach, and we suggest that Poincar\'e transformations for special
relativistic systems can be understood as residual gauge transformations. In
Appendices, we give the general computation of the equations of motion and the
Lagrangian for any Bianchi-Type vacuum metric and for spatially homogeneous
Maxwell fields in a nondynamical background (with zero currents). We also
illustrate our counting of degrees of freedom in an Appendix.
