Also regulators and cycle maps.
Why do CTs in AG have twists? One explanation (given by Scholl in some conversation) involves cycle maps. This should be related to weights.
Gillet: Riemann-Roch theorems… constructs characteristic classes from K-theory to a quite general cohomology theory. See also Feliu, page 84.
nLab page on D90 Char classes Regulators and Cycle maps