  We present a comprehensive study on discrete morphological symmetries of
dynamical systems, which are commonly observed in biological and artificial
locomoting systems, such as legged, swimming, and flying animals/robots/virtual
characters. These symmetries arise from the presence of one or more planes/axis
of symmetry in the system's morphology, resulting in harmonious duplication and
distribution of body parts. Significantly, we characterize how morphological
symmetries extend to symmetries in the system's dynamics, optimal control
policies, and in all proprioceptive and exteroceptive measurements related to
the system's dynamics evolution. In the context of data-driven methods,
symmetry represents an inductive bias that justifies the use of data
augmentation or symmetric function approximators. To tackle this, we present a
theoretical and practical framework for identifying the system's morphological
symmetry group $\G$ and characterizing the symmetries in proprioceptive and
exteroceptive data measurements. We then exploit these symmetries using data
augmentation and $\G$-equivariant neural networks. Our experiments on both
synthetic and real-world applications provide empirical evidence of the
advantageous outcomes resulting from the exploitation of these symmetries,
including improved sample efficiency, enhanced generalization, and reduction of
trainable parameters.
