  We investigate possible generalizations of the de Finetti theorem to bi-free
probability. We first introduce a twisted action of the quantum permutation
groups corresponding to the combinatorics of bi-freeness. We then study
properties of families of pairs of variables which are invariant under this
action, both in the bi-noncommutative setting and in the usual noncommutative
setting. We do not have a completely satisfying analogue of the de Finetti
theorem, but we have partial results leading the way. We end with suggestions
concerning the symmetries of a potential notion of n-freeness.
