A total Steenrod operation as homomorphism of Steenrod algebra-modules

Let p be an odd prime, M be an unstable locally finite module over the Steenrod algebra, and let Phi(S) be the localization out of the Euler class of the mod-p cohomology ring of the group Z/p. We prove that the Singer evaluation map d : Phi(GL1)(S) circle times M -> M is dually related to a total operation chi(P) : M -> Phi(GL1)(S) circle times M. We determine the exotic A(p)-module structure on the target which makes chi(P) an A(p)-linear map, give a new proof of the Adem relations and find some new identities involving the Bockstein and the pth reduced powers.