In this talk I will describe how comparing different constructions of cyclic cocycles for complete symbols of pseudodifferential operators leads to interesting index-theoretic consequences. As a consequence of these results, one can define pairings of Alexnder-Spanier cocycles with the algebraic K-theory of the algebra of complete symbols of bounded pseudodifferential operators. As an illustration, we explicitly compute these secondary pairings on the higher Loday symbols in algebraic K-theory.

This is a joint work with H. Moscovici.