  We show that the Killing spinor equations of all supergravity theories which
may include higher order corrections on a (r,s)-signature spacetime are
associated with twisted covariant form hierarchies. These hierarchies are
characterized by a connection on the space of forms which may not be degree
preserving. As a consequence we demonstrate that the form Killing spinor
bi-linears of all supersymmetric backgrounds satisfy a suitable generalization
of conformal Killing-Yano equation with respect to this connection. To
illustrate the general proof the twisted covariant form hierarchies of some
supergravity theories in 4, 5, 6, 10 and 11 dimensions are also presented.
