For any finite-dimensional *-algebra, one can choose a basis such that the coefficients tensor of the anti-linear map $(a,b)\rightarrow (ab)^*$ becomes cyclically symmetric. (Any *-algebra is isomorphic to a direct sum of full matrix algebras which has this property for the basis given by the matrix entries.)
For finite Hopf C*-algebras (aka finite Kac algebras) it is known that the co-algebra can also be made a *-(co-)algebra (with the adjoint operation given by the product of the antipode and the adjoint operation of the algebra; on the level of the coefficient tensors the axioms for algebra and co-algebra are identical). So one can also choose a basis such that the coefficients tensor for $\Delta \circ S \circ *$ becomes cyclically symmetric.
Now, can one always choose a basis so that both tensors become cyclically symmetric simultanously? All Kac algebras I'm aware of (e.g. all group algebras) have this property.
If yes, I'd also be interested in the same question for finite involutive weak Hopf C*-algebras.
