If $H$ is a Hopf algebra over a field $\mathbb{k}$, then its category of left modules $\operatorname{\mathsf{Rep}} H$ is monoidal, and furthermore left rigid (admits all left duals, as a consequence of the antipode). If the antipode is invertible, then $\operatorname{\mathsf{Rep}} H$ admits all right duals as well.
Now suppose that we have some pivotal structure on $\operatorname{\mathsf{Rep}} H$, i.e. natural isomorphisms $i_A: A \to A^{**}$ such that $i_{A \otimes B} = i_A \otimes i_B$: this is strictly more data than $\operatorname{\mathsf{Rep}} H$. What does this extra data correspond to, in terms of $H$ itself? I'm looking for an answer along the same lines as "a braiding on $\operatorname{\mathsf{Rep}} H$ is an $R$-matrix", or "a twist on $\operatorname{\mathsf{Rep}} H$ is an element of $H$ such that ...". I'm having trouble finding this written down anywhere without the assumption that $\operatorname{\mathsf{Rep}} H$ is braided, i.e. $H$ is quasitriangular.
I have found something for a finite dimensional Hopf algebra. The corresponding datum is called pivot.
A pivot is a group like element $g \in H$ (i.e $\Delta(g)= g \otimes g$) such that $S^2(x)=g x g^{-1}$, $\forall x \in H$.
We call $(H,g)$ a pivotal Hopf algebra. (It depends on the choice of the pivot.)
If $(H,g)$ is a pivotal Hopf algebra, then $\mathsf{Rep} \ H$ is pivotal, with natural isomorphisms $i_A : A \to A^{\star \star}$, $v \mapsto g \circ \delta(v)$ where $\delta(v)= <.,v>$ ($<.,.>$ is the usual pairing of $A^{\star} \otimes A$).
You can find much more details in the section 4 of this article from A.Beliakova, C.Blanchet, A. Gainutdinov Modified trace is a symmetrised integral.
It is done for finite dimensional Hopf algebra, but I guess you can generalise the pivot and the fact that it induce a pivotal structure to any Hopf algebra. Or is there some obstructions I do not see?