  We observe that the Poincare duality isomorphism for a string manifold is an
isomorphism of modules over the subalgebra A(2) of the modulo 2 Steenrod
algebra. In particular, the pattern of the operations Sq^1, Sq^2, and Sq^4 on
the cohomology of a string manifold has a symmetry around the middle dimension.
We characterize this kind of cohomology operation duality in term of the
annihilator of the Thom class of the negative tangent bundle, and in terms of
the vanishing of top-degree cohomology operations. We also indicate how the
existence of such an operation-preserving duality implies the integrality of
certain polynomials in the Pontryagin classes of the manifold.
