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A bidirectional transformation (BX) keeps a pair of interrelated models synchronized. Symmetric BXs are those for which neither model in the pair fully determines the other. We build two algebraic frameworks for symmetric BXs, with one correctly implementing the other, and both being delta-based generalizations of known state-based frameworks. We identify two new algebraic laws-weak undoability and weak invertibility, which capture important semantics of BX and are useful for both state- and delta-based settings. Our approach also provides a flexible tool architecture adaptable to different user’s needs.