  Infinitesimal symmetries of a partial differential equation (PDE) can be
defined algebraically as the solutions of the linearization (Frechet
derivative) equation holding on the space of solutions to the PDE, and they are
well-known to comprise a linear space having the structure of a Lie
algebra.Solutions of the adjoint linearization equation holding on the space of
solutions to the PDE are called adjoint-symmetries. Their algebraic structure
for general PDE systems is studied herein. This is motivated by the
correspondence between variational symmetries and conservation laws arising
from Noether's theorem, which has a well-known generalization to
non-variational PDEs, where infinitesimal symmetries are replaced by
adjoint-symmetries, and variational symmetries are replaced by multipliers
(adjoint-symmetries satisfying a certain Euler-Lagrange condition). Several
main results are obtained. Symmetries are shown to have three different linear
actions on the linear space of adjoint-symmetries. These linear actions are
used to construct bilinear adjoint-symmetry brackets, one of which is a
pull-back of the symmetry commutator bracket and has the properties of a Lie
bracket. The brackets do not use or require the existence of any local
variational structure (Hamiltonian or Lagrangian) and thus apply to general PDE
systems. One of the symmetry actions is shown to encode a pre-sympletic
(Noether) operator and a symplectic 2-form, which lead to the construction of a
symplectic 2-form and Poisson bracket for evolution systems. The generalized
KdV equation in potential form is used to illustrate all of the results.
