  The collective dynamics of mobile scatterers and light in optical resonators
generates complex behaviour. For strong transverse illumination a phase
transition from homogeneous to crystalline particle order appears. In contrast,
a gas inside a single-side pumped ring cavity exhibits an instability towards
bunching and collective acceleration called collective atomic recoil lasing
(CARL). We demonstrate that by driving two orthogonally polarized counter
propagating modes of a ring resonator one realises both cases within one
system. The corresponding phase diagram depending on the two pump intensities
exhibits regions in which either a generalized form of self-ordering towards a
travelling density wave with constant centre of mass velocity or a CARL
instability is formed. Controlling the cavity driving then allows to accelerate
or slow down and trap a sufficiently dense beam of linearly polarizable
particles.
