  We extensively study the evolution and distinct signatures of cosmological
models, in which dark energy interacts directly with dark matter. We first
focus on the imprints of these coupled models on the cosmic microwave
background temperature power spectrum, in which we discuss the multipole peak
separation together with the integrated Sachs-Wolfe effect. We also address the
growth of matter perturbations, and disentangle the interacting dark energy
models using the expansion history together with the growth history. We find
that a disformal coupling between dark matter and dark energy induces
intermediate-scales and time-dependent damped oscillatory features in the
matter growth rate function, a unique characteristic of this coupling. Apart
from the disformal coupling, we also consider conformally coupled models,
together with models which simultaneously make use of both couplings.
