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The Leibniz-Hopf algebra is the free associative algebra with one generator in each degree and co-product given by the Cartan formula. The mod 2 Steenrod algebra naturally occurs as a quotient of this Hopf algebra by the Adem relations. Several years ago, motivated by its link with commutativity of ring spectra, Martin Crossley and Sarah Whitehouse attempted to compute the invariants under conjugation in the dual Steenrod algebra. A complete calculation was not possible then, so we present a different approach to this problem. In particular, we calculate the conjugation invariants in the mod 2 dual Leibniz-Hopf algebra. Since the dual Steenrod algebra is a sub Hopf algebra of this algebra, so the conjugation invariants in the dual Steenrod algebra are the intersection of the Leibniz-Hopf conjugation invariants and the dual Steenrod algebra. We show that, like in the dual Steenrod algebra, the degree-wise dimension of the conjugation invariants is ''approximately'' half that of the whole algebra.