Summary: We provide a geometric interpretation of generalized Jacobi morphisms in the framework of finite order variational sequences. \par Jacobi morphisms arise classically as an outcome of an invariant decomposition of the second variation of a Lagrangian. Here they are characterized in the context of generalized Lagrangian symmetries in terms of variational Lie derivatives of generalized Euler-Lagrange morphisms. We introduce the variational vertical derivative and stress its link with the classical concept of variation. The relation with generalized Helmholtz morphisms is also clarified.