Ben, I do not know anything on the subject, but another example of this kind is the duality between quantifiers $\exists$ and $\forall$, and more generally, you will find a lot of such examples in almost any interesting fibration (with comonad $f^* \circ \Pi_f$ and monad $f^* \circ \Sigma_f$). However, I do not understand how such a monad-comonad pair could form a Frobenius algebra, as the later requires both monoidal and comonoidal structure to operate on a single carrier, and in our examples the carriers are just far different (i.e. the functors are different). –  Michal R. Przybylek May 27 '12 at 12:55
Any identity functor is both a monad and a comonad. What I was to say, is that in our examples this is not the case. I still do not know what is that mysterious "interaction" between monads and comonads that you are refering to. However, it is easy to check, that if a monad/comonad pair arises from an adjoint triple $\Sigma_f \dashv f \dashv \prod_f$ and $\Sigma_f \approx \prod_f$ then the pair form a frobenius algebra. –  Michal R. Przybylek May 28 '12 at 11:13