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