Actually it is Hermitian conjugation in both senses: Both wrt. Dirac indices and as differential operators. In particular we still have to integrate by parts. Also note that the complex conjugate of a product two supernumbers $z,w$ is by definition the opposite order: $(z w)^{\ast}=w^{\ast} z^{\ast}$.

Ultimately, it is Hermitian conjugation in both senses: Both wrt. Dirac indices and as differential operators. In particular we still have to integrate by parts. Also note that the complex conjugate of a product two supernumbers $z,w$ is by definition the opposite order: $(z w)^{\ast}=w^{\ast} z^{\ast}$.