  We introduce a new class of non-local games, and corresponding densities,
which we call bisynchronous. Bisynchronous games are a subclass of synchronous
games and exhibit many interesting symmetries when the algebra of the game is
considered. We develop a close connection between these non-local games and the
theory of quantum groups which recently surfaced in studies of graph
isomorphism games. When the number of inputs is equal to the number of outputs,
we prove that a bisynchronous density arises from a trace on the quantum
permutation group.
  Each bisynchronous density gives rise to a completely positive map and we
prove that these maps are factorizable maps.
