We consider quasilinear parabolic equations on $\mathbb R^N$
satisfying certain symmetry conditions. We prove that
bounded positive solutions decaying to zero at
spatial infinity are asymptotically radially symmetric about a
center.  The asymptotic center of symmetry is not fixed a priori
(and depends on the solution) but it is independent of time.
We also prove a  similar theorem on reflectional symmetry.