In this question, bimonadic category is a category $C$ such that $C$ and $C^{\text{op}}$ are monadic over $\mathrm{Set}$.

How many bimonadic categories are there? Can we classify them all?

Currently (updated):

1. 1, $\mathrm{Set}$, $\mathrm{CompHaus}$, $\mathrm{SupLat}$ are bimonadic.
2. $C \times D$ is bimonadic if $C$ and $D$ are bimonadic.

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. $\operatorname{Hom}(P, -)$ faithful)
2. $P$ is projective (that is, $\operatorname{Hom}(P, -)$ preserves epimorphisms)
3. $P$ has all copowers $\coprod_A P$
4. For any $X \in \operatorname{Ob} C$, the natural morphism $\coprod_{f: P \to X} P \to X$ is a regular epimorphism.

Comonadic cogenerator is a formal dualization of this concept.

This characterization is not an answer to the question, because directly from it it is not clear how to check whether a given category is bimonadic.

P.S. A similar question Can the opposite of an elementary topos be an elementary topos? about toposes states that the opposite category of a locally presentable category is never locally presentable (with the exception of complete posets?), but a monadic category is not necessarily locally presentable (for example, the category of frames).