  We derive necessary and sufficient conditions for arbitrary multi--mode (pure
or mixed) Gaussian states to be equivalent under Gaussian local unitary
operations. To do so, we introduce a standard form for Gaussian states, which
has the properties that (i) every state can be transformed into its standard
form via Gaussian local unitaries and (ii) it is unique and (iii) it can be
easily computed. Thus, two states are equivalent under Gaussian local unitaries
iff their standard form coincides. We explicitly derive the standard form for
two-- and three--mode Gaussian pure states. We then investigate transformations
between these classes by means of Gaussian local operations assisted by
classical communication. For three-mode pure states, we identify a global
property that cannot be created but only destroyed by local operations. This
implies that the highly entangled family of symmetric three--mode Gaussian
states is not sufficient to generated all three-mode Gaussian states by local
Gaussian operations.
