  A geometrical formulation for adjoint-symmetries as 1-forms is studied for
general partial differential equations (PDEs), which provides a dual
counterpart of the geometrical meaning of symmetries as tangent vector fields
on the solution space of a PDE. Two applications of this formulation are
presented. Additionally, for systems of evolution equations, adjoint-symmetries
are shown to have another geometrical formulation given by 1-forms that are
invariant under the flow generated by the system on the solution space. This
result is generalized to systems of evolution equations with spatial
constraints, where adjoint-symmetry 1-forms are shown to be invariant up to a
functional multiplier of a normal 1-form associated to the constraint
equations. All of the results are applicable to the PDE systems of interest in
applied mathematics and mathematical physics.
