  General Relativity is usually formulated as a theory with gauge invariance
under the diffeomorphism group, but there is a 'dilaton' formulation where it
is in addition invariant under Weyl transformations, and a 'unimodular'
formulation where it is only invariant under the smaller group of special
diffeomorphisms. Other formulations with the same number of gauge generators,
but a different gauge algebra, also exist. These different formulations provide
examples of what we call 'inessential gauge invariance', 'symmetry trading' and
'linking theories'; they are locally equivalent, but may differ when global
properties of the solutions are considered. We discuss these notions in the
Lagrangian and Hamiltonian formalism.
