  Local actions of $\mathbb{P}_\mathbb{N}$, the group of finite permutations on
$\mathbb{N}$, on quasi-local algebras are defined and proved to be
$\mathbb{P}_\mathbb{N}$-abelian. It turns out that invariant states under local
actions are automatically even, and extreme invariant states are strongly
clustering. Tail algebras of invariant states are shown to obey a form of the
Hewitt and Savage theorem, in that they coincide with the fixed-point von
Neumann algebra. Infinite graded tensor products of $C^*$-algebras, which
include the CAR algebra, are then addressed as particular examples of
quasi-local algebras acted upon $\mathbb{P}_\mathbb{N}$ in a natural way.
Extreme invariant states are characterized as infinite products of a single
even state, and a de Finetti theorem is established. Finally, infinite products
of factorial even states are shown to be factorial by applying a twisted
version of the tensor product commutation theorem, which is also derived here.
