  Over a field of characteristic zero, we introduce two motivic operations on
additive higher Chow cycles: analogues of the Connes boundary $B$ operator and
the shuffle product on Hochschild complexes. The former allows us to apply the
formalism of mixed complexes to additive Chow complexes building a bridge
between additive higher Chow theory and additive $K$-theory. The latter induces
a wedge product on additive Chow groups for which we show that the Connes
operator is a graded derivation for the wedge product using a variation of a
Totaro's cycle. Hence, the additive higher Chow groups with the wedge product
and the Connes operator form a commutative differential graded algebra. On
zero-cycles, they induce the wedge product and the exterior derivation on the
absolute K\"ahler differentials, answering a question of S. Bloch and H.
Esnault.
