  We study the non-equilibrium dynamics of Abelian anyons in a one-dimensional
system. We find that the interplay of anyonic statistics and interactions gives
rise to spatially asymmetric particle transport together with a novel dynamical
symmetry that depends on the anyonic statistical angle and the sign of
interactions. Moreover, we show that anyonic statistics induces asymmetric
spreading of quantum information, characterized by asymmetric light cones of
out-of-time-ordered correlators. Such asymmetric dynamics is in sharp contrast
with the dynamics of conventional fermions or bosons, where both the transport
and information dynamics are spatially symmetric. We further discuss
experiments with cold atoms where the predicted phenomena can be observed using
state-of-the-art technologies. Our results pave the way toward experimentally
probing anyonic statistics through non-equilibrium dynamics.
