Doubly strongly first-order dependencies

Team Semantics is a generalization of Tarskian Semantics in which formulas are satisfied with respect to sets of assignments, called teams. In this semantics, it is possible to add new atoms and connectives expressing dependencies between possible values of variables. Some of the resulting logics are more expressive than first-order logic while others are not. I study the (relativizable) atoms and families of atoms that do not increase the expressive power of first-order logic when they and their complements are added to it, separately or jointly, calling them doubly strongly first-order dependencies and finding a characterization for them.