  We study bipartite unitary operators which stay invariant under the local
actions of diagonal unitary and orthogonal groups. We investigate structural
properties of these operators, arguing that the diagonal symmetry makes them
suitable for analytical study. As a first application, we construct large new
families of dual unitary gates in arbitrary finite dimensions, which are
important toy models for entanglement spreading in quantum circuits. We then
analyze the non-local nature of these invariant operators, both in discrete
(operator Schmidt rank) and continuous (entangling power) settings. Our
scrutiny reveals that these operators can be used to simulate any bipartite
unitary gate via stochastic local operations and classical communication.
Furthermore, we establish a one-to-one connection between the set of local
diagonal unitary invariant dual unitary operators with maximum entangling power
and the set of complex Hadamard matrices. Finally, we discuss
distinguishability of unitary operators in the setting of the stated diagonal
symmetry.
