  In this paper we show the existence of an action of Chow correspondences on
the cohomology of reciprocity sheaves. In order to do so, we prove a number of
structural results, such as a projective bundle formula, a blow-up formula, a
Gysin sequence, and the existence of proper pushforward. In this way we recover
and generalize analogous statements for the cohomology of Hodge sheaves and
Hodge-Witt sheaves.
  We give several applications of the general theory to problems which have
been classically studied. Among these applications, we construct new birational
invariants of smooth projective varieties and obstructions to the existence of
zero-cycles of degree one from the cohomology of reciprocity sheaves.
