  When animal groups move coherently in the form of a flock, their trajectories
are not all parallel, the individuals exchange their position in the group. In
this Letter we introduce a measure of this mixing dynamics, which we quantify
as the winding of the braid formed from the particle trajectories. Building on
a paradigmatic flocking model we numerically and theoretically explain the
winding statistics, and show that it is predominantly set by the global twist
of the trajectories as a consequence of a spontaneous symmetry breaking.
