  In this paper, we introduce and study a cyclic cohomology theory
$H^\bullet_\lambda(A,C,\psi)$ for an entwining structure $(A,C,\psi)$ over a
field $k$. We then provide a complete description of the cocycles and the
coboundaries in this theory using entwined traces applied to dg-entwining
structures over $(A,C,\psi)$. We then apply these descriptions to construct a
pairing $ H^m_\lambda(A,C,\psi) \otimes H^n_\lambda(A',C',\psi')
\longrightarrow H^{m+n}_\lambda(A \otimes A', C \otimes C', \psi \otimes \psi')
$, where $(A,C,\psi)$ and $(A',C',\psi')$ are entwining structures.
