  We show that the breaking of the rotation symmetry of spiral waves in
two-dimensional complex (period-doubled or chaotic) oscillatory media by
synchronization defect lines (SDL) is accompanied by an intrinsic drift of the
pattern. Single vortex motion changes from ballistic flights at a well-defined
angle from the SDL to Brownian-like diffusion when the turbulent character of
the medium increases. It gives rise, in non-turbulent multi-spiral regimes, to
a novel ``vortex liquid''.
