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We consider quantum integrable models solvable by the nested algebraic Betheansatz and possessing $\mathfrak{gl}(N)$-invariant $R$-matrix. We study twotypes of Bethe vectors. The first type corresponds to the original monodromymatrix. The second type is associated to a monodromy matrix closely related tothe inverse of the monodromy matrix. We show that these two types of the Bethevectors are identical up to normalization and reshuffling of the Betheparameters. To prove this correspondence we use the current approach. Thisidentity gives new combinatorial relations for the scalar products of the Bethevectors.

New symmetries of $\mathfrak{gl}(N)$-invariant Bethe vectors