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The reversing symmetry group is considered in the setting of symbolic dynamics. While this group is generally too big to be analysed in detail, there are interesting cases with some form of rigidity where one can determine all symmetries and reversing symmetries explicitly. They include Sturmian shifts as well as classic examples such as the Thue--Morse system with various generalisations or the Rudin--Shapiro system. We also look at generalisations of the reversing symmetry group toarXiv:1611.05756v2 fatcat:nxeafvvlyzhj5n6o2uqojemucu