In this paper we investigate the behavior of mutation-based evolutionary algorithms on highly symmetric binary constraint satisfaction problems. With empirical methods we study why and when these algorithms perform better under the stepwise adaptive weighting of penalties (SAWing) than under the standard penalty function. We observe that SAWing has little effect when the local optima of the symmetric problems are not very strong. However, while the use of the standard penalty function can lead to strong local optima, the SAWing mechanism can avoid this situation. The symmetric problems we consider are the standard one-dimensional Ising model and a more complex construction with the Ising model as the core component.