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We study bipartite unitary operators which stay invariant under the localactions of diagonal unitary and orthogonal groups. We investigate structuralproperties of these operators, arguing that the diagonal symmetry makes themsuitable for analytical study. As a first application, we construct large newfamilies of dual unitary gates in arbitrary finite dimensions, which areimportant toy models for entanglement spreading in quantum circuits. We thenanalyze the non-local nature of these invariant operators, both in discrete(operator Schmidt rank) and continuous (entangling power) settings. Ourscrutiny reveals that these operators can be used to simulate any bipartiteunitary gate via stochastic local operations and classical communication.Furthermore, we establish a one-to-one connection between the set of localdiagonal unitary invariant dual unitary operators with maximum entangling powerand the set of complex Hadamard matrices. Finally, we discussdistinguishability of unitary operators in the setting of the stated diagonalsymmetry.

Diagonal unitary and orthogonal symmetries in quantum theory II: Evolution operators