Let $F$ be a monad on some category $\mathsf{C}$ and $G$ be a comonad on the same category. Assume further that they "commute" (see below): $FG \cong GF$. Then, for lack of a better name, one can define "$(F,G)$-bimodules": objects $X$ equipped with an $F$-algebra structure and a $G$-coalgebra structure, both compatible in the sense that the two following maps are equal: $$(F(X) \to X \to G(X)) = (F(X) \to G(F(X)) \to G(X)).$$
Is this kind of structure studied? What kind of properties does it have?
For the precise meaning of "$F$ and $G$ commute", consider the case where $\mathsf{C}$ is the category of modules over some commutative ring, you have an algebra $A$, a coalgebra $C$, and $F = - \otimes A$ and $G = C \otimes -$ are respectively the "free right $A$-module" and the "cofree left $C$-comodule" functors. Then there is a natural isomorphism $C \otimes (M \otimes A) \cong (C \otimes M) \otimes A$, which is moreover compatible with the monad and comonad structure maps in a way that I don't know how to formalize precisely.
A distributive law of a monad $(F, \eta, \mu)$ over a comonad $(G, \epsilon, \delta)$ is a natural transformation $\xi : F G \Rightarrow G F$ that satisfies the following equations:
\begin{align} \xi \bullet \eta G & = G \eta & \xi \bullet \mu G & = G \mu \bullet \xi F \bullet F \xi \\ \epsilon F \bullet \xi & = F \epsilon & \delta F \bullet \xi & = G \xi \bullet \xi G \bullet F \delta \end{align}
For example, in a monoidal category, if $(F, \eta, \mu)$ is the monad induced by a monoid and $(G, \epsilon, \delta)$ is the comonad induced by a comonoid, then the associator defines a distributive law in the sense above.
Now, given a distributive law as above, a bialgebra is an object $B$ with an $(F, \eta, \mu)$-action $\alpha : F B \to B$ and a $(G, \epsilon, \delta)$-coaction $\beta : B \to G B$ that satisfies the following equation:
$$\beta \circ \alpha = G \alpha \circ \xi_B \circ F \beta$$
This is studied in more detail in [Power and Watanabe, Combining a monad and a comonad].