  We study monotone operators in reflexive Banach spaces that are invariant
with respect to a group of suitable isometric isomorphisms and we show that
they always admit a maximal extension which preserves the same invariance. A
similar result applies to Lipschitz maps in Hilbert spaces, thus providing an
invariant version of Kirzsbraun-Valentine extension Theorem. We then provide a
relevant application to the case of monotone operators in $L^p$-spaces of
random variables which are invariant with respect to measure-preserving
isomorphisms, proving that they always admit maximal dissipative extensions
which are still invariant by measure-preserving isomorphisms. We also show that
such operators are law invariant, a much stronger property which is also
inherited by their resolvents, the Moreau-Yosida approximations, and the
associated semigroup of contractions. These results combine explicit
representation formulae for the maximal extension of a monotone operator based
on selfdual lagrangians and a refined study of measure-preserving maps in
standard Borel spaces endowed with a nonatomic measure, with applications to
the approximation of arbitrary couplings between measures by sequences of maps.
