  We study natural invariance properties of functionals defined on L\'evy
processes and show that they can be described by a simplified structure of the
deterministic chaos kernels in It\^o's chaos expansion. These structural
properties of the kernels relate intrinsically to a measurability with respect
to invariant $\sigma$-algebras. This makes it possible to apply deterministic
functions to invariant functionals on L\'evy processes while keeping the
simplified structure of the kernels. This stability is crucial for
applications. Examples are given as well.
