Resumen
For strongly monotone skew-product semiflows on which a compact connected group acts, it is shown that any stable minimal set is residually symmetric and any uniformly stable trajectory is asymptotically symmetric. These results are then applied to study the spatio-temporal asymptotics of stable solutions of reaction-diffusion equations on a symmetric domain in time-recurrent structures including almost periodicity. In particular, the 1-covering property of omega-limit sets is established for uniformly stable bounded solutions of reaction-diffusion equations on a ball.