  Symmetries play a conspicuous role in the large-scale behavior of critical
systems. While in equilibrium they allow to classify asymptotics into different
universality classes, out of equilibrium they can emerge, some times
unexpectedly, as collective properties which are not explicit in the 'bare'
interactions. Here we elucidate the emergence of an up-down symmetry in the
asymptotic behavior of the stochastic scalar Burgers equation in one and two
dimensions, manifested by the occurrence of Gaussian fluctuations for the
physical field. This robustness of Gaussian behavior contradicts naive
expectations, due to the detailed relation ---including the same set of
symmetries--- between Burgers equation and the Kardar-Parisi-Zhang equation,
which paradigmatically displays non-Gaussian fluctuations described by
Tracy-Widom distributions. We reach our conclusions via a dynamic
renormalization group study of the field statistics, confirmed by direct
evaluation of the field probability distribution function from numerical
simulations of the dynamical equation. All odd-order cumulants cancel exactly,
while the excess kurtosis vanishes for large systems, indeed consistent with
Gaussian behavior.
