We carry out a systematic analysis of the correlation between similarity of protein three-dimensional structures and their evolutionary relationships. The structural similarity is quantitatively identified by an all-against-all comparison of the spatial arrangement of secondary structural elements in nonredundant 967 representative proteins, and the evolutionary relationship is judged according to the definition of superfamily in the SCOP database. We find the following symmetry rule: a protein pair that has similar folds but belong to different superfamilies has (with a very rare exception) certain internal symmetry in its common similar folds. Possible reasons behind the symmetry rule are discussed.