Symmetry properties are presented for a multidimensional dispersion functional. If the system of linearized Vlasov‐field equations is ‘‘completely Hamiltonian,’’ the dispersion operator satisfies a certain formal self‐adjointness property as a function of omega. For appropriate boundary conditions this implies a relation between an eigenfunction and its dual. If the equilibrium admits ‘‘conjugate orbits’’ for a completely Hamiltonian system and if the ‘‘conjugate‐orbit parity condition’’ is satisfied, then the kinetic part of the dispersion matrix is symmetric. For this case and for appropriate boundary conditions the entire dispersion matrix for the multispecies Vlasov or Vlasov‐fluid models is symmetric. It then follows that the complex conjugate of the dual eigenfunction is proportional to the eigenfunction itself. The analytic continuation of the dispersion functional of the linearized Vlasov‐field equations into the lower half of the frequency plane is derived.