The group algebra $k(G)$ of any group $G$ satisfies as a Hopf algebra the following identities: $$ S\otimes S\circ \varDelta=\sigma\circ\Delta\circ S $$ $$ \nabla\circ S\otimes S=S\circ\nabla\circ\sigma $$ where $S$ is the antipode, $\Delta$, the comultiplication, $\nabla$, the multiplication, and $\sigma:x\otimes y\mapsto y\otimes x$.
Is this valid for all Hopf algebras or only for some special class?
EDIT. I asked this question later at MathOverflow.
Your first identity says that $S$ is a coalgebra anti-homomorphism, and your second says $S$ is an algebra anti-homomorphism. Both of these are true for all Hopf algebras, see for example http://library.msri.org/books/Book43/files/nik.pdf or http://ncatlab.org/nlab/show/Hopf+algebra, or one of the books by Sweedler or Abe.