We should call them “dual operators” not “anti-operators” because otherwise the dual commutator is called the anti-commutator, which can be confused with the anti-commutator, and the dual anti-commutator being called the anti-anti-commutator can be confused with the commutator.
I think it helps to have them all in the algebra, for completeness, and possible unknown applications. This does not affect the correctness of enki’s arguments.