A monadic category is a category of algebras over a monad in $\mathrm{Set}$ (I'm not sure if this is standard terminology). Can we describe all monadic categories whose opposite categories are monadic? I see only one example: the terminal category. A similar question 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 for $C$ categories 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. $P$ 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..