Adler's principle and the requirement of algebraic duality are discussed with relation to individual terms of the expansion of the $n$-point dual amplitude with respect to homogeneous functions of degree $r=1,2,\dots$ of the kinematic invariants $s_{ik}$. The fulfillment of Adler's principle is ensured by the use of a phenomenological Lagrangian that is invariant under the considered symmetry group and contains arbitrarily many derivatives of the meson fields. It is shown that the requirement of algebraic duality leads to more or less strict restrictions depending on the structure of the symmetry group.