A unified matrix treatment which is defined for binary Walsh matrices is presented. This unified treatment, based on the concept of dyadic symmetry, defines Walsh matrices of different orderings using a simple equation. This equation in turn provides a straightforward derivation of various reordering schemes and Walsh matrix properties. Various fast computational algorithms can also be derived within the same framework using dyadic decompositions. It is hoped that this unified treatment of Walsh matrices using dyadic symmetry will provide a better understanding of Walsh transforms, and will lead to the discovery of more useful transforms.