  We analyze carefully the problem of gauge symmetries for Bianchi models, from
both the geometrical and dynamical points of view. Some of the geometrical
definitions of gauge symmetries (=``homogeneity preserving diffeomorphisms'')
given in the literature do not incorporate the crucial feature that local gauge
transformations should be independent at each point of the manifold of the
independent variables ( = time for Bianchi models), i.e, should be arbitrarily
localizable ( in time). We give a geometrical definition of homogeneity
preserving diffeomorphisms that does not possess this shortcoming. The proposed
definition has the futher advantage of coinciding with the dynamical definition
based on the invariance of the action ( in Lagrangian or Hamiltonian form). We
explicitly verify the equivalence of the Lagrangian covariant phase space with
the Hamiltonian reduced phase space. Remarks on the use of the Ashtekar
variables in Bianchi models are also given.
