  In $1998$, Connes and Moscovici defined the cyclic cohomology of Hopf
algebras. In $2010$, Khalkhali and Pourkia proposed a braided generalization:
to any Hopf algebra $H$ in a braided category $\mathcal B$, they associate a
paracocyclic object in $\mathcal B$. In this paper we explicitly compute the
powers of the paracocyclic operator of this paracocyclic object. Also, we
introduce twisted modular pairs in involution for $H$ and derive (co)cyclic
modules from them. Finally, we relate the paracocyclic object associated with
$H$ to that associated with an $H$-module coalgebra via a categorical version
of the Connes-Moscovici trace.
