In this work, we exploit geometric symmetries in extended objects in order to improve Bayesian tracking algorithms that use Spatial Distribution Models, Greedy Association Models as used in curve fitting, and Random Hypersurface Models. The key idea is to describe symmetric objects by solely modeling the non-redundant part of the shape, while the remainder of the shape follows from symmetry. Following this idea, we develop simplified versions for the three models that take advantage of the symmetry. Exploiting symmetries yields two major benefits. First, complex symmetric shapes can be equivalently represented by a fraction of the original shape parameters. Second, when using sample-based filters, such as the widely used Unscented Kalman Filter, symmetry yields a higher effective sample resolution. It is worth mentioning that estimating even simple objects such as a stick, which only have one reflectional symmetry, can be significantly improved.