We consider nonautonomous quasilinear parabolic equations satisfying certain symmetry
conditions. We prove that each positive bounded  solution  $u$ on  $\mathbb R^N\times (-\infty,T)$ decaying
to zero at spatial infinity uniformly with respect to time is radially symmetric  around some origin in  $\mathbb R^N$.
The origin  depends on the solution but is independent of time. We also consider the linearized equation
along   $u$ and prove that each bounded (positive or not) solution is a linear combination of a radially symmetric
solution and (nonsymmetric) spatial derivatives of  $u$. Theorems on  reflectional symmetry are also given.