In this question, a monadic category is a category of algebras over a monad in $\mathrm{Set}$.
> Can we describe all monadic categories whose opposite categories are monadic? 

I see only one example: the terminal category (UPD: two more excellent non-trivial examples have been given in the comments so far). A similar [question](https://mathoverflow.net/questions/390037/can-the-opposite-of-an-elementary-topos-be-an-elementary-topos?rq=1) about toposes states that the opposite category of a locally representable category is never locally representable (with the exception of $1$), but a monadic category is not necessarily locally representable (for example, the category of frames).

By the characterization theorem for monadic categories over $\mathrm{Set}$ (see Borceux, HCA II), this question is equivalent to searching categories $C$ with the following properties
1) $C$ Barr-exact and co(Barr-exact).
2) $C$ has a monadic generator and a comonadic cogenerator.

A monadic generator is an object $P$ such that
1) $P$ is a separator (i.e. $\mathrm{Hom}(P, -)$ faithful)
2) $P$ is projective (that is, $\mathrm{Hom}(P, -)$ preserves epimorphisms)
3) $P$ has all copowered $\coprod_A P$
4) For any $X \in \mathrm{Ob} C$, the natural morphism $\coprod_{f: P \to X} P \to X$ is a regular epimorphism.

Comonadic cogenerator is a formal dualization of this concept..