In a variety of one-dimensional nonequilibrium systems, there exist uniform states that may undergo bifurcations to spatially periodic states. The long-wavelength dynamics of these spatial patterns, such as an array of convective rolls or a driven interface between two thermodynamic phases, can often be derived on the basis of the symmetries of the physical system. Secondary bifurcations of these patterns may be associated with the subsequent loss of remaining symmetries. Here, we study the transition that results from the loss of reflection symmetry (parity) and show that observations made in several recent experiments appear to be signatures of this bifurcation. Most of the common features seen in the disparate experiments follow from the simplest Ginzburg-Landau equations covariant under the remaining symmetries. It is shown that nucleated localized regions of broken parity travel in a direction determined by the sense of the asymmetry, and the passage of a localized inclusion of broken parity leads to a change in the wavelength of the underlying modulated state, and leaves the system closer to an invariant wavelength. Such behavior is in close correspondence with the properties of ‘‘solitary modes’’seen in experiment. When a system supports extended regions of broken parity, a boundary between those of opposite parity can be considered as a source or sink of asymmetric cells and a ‘‘spatiotemporal grain boundary.’’The creation or destruction of cells at the interface is reminiscent of ‘‘phase slip centers’’in one-dimensional superconductors. The simplest dynamics consistent with the symmetries are identical to those of the time-dependent Ginzburg-Landau equation for a superconductor in an applied electric field. The experimental observation of approximate length subtraction of colliding regions of broken parity follows from this analogy.