Is there a "Coalgebra - Cogeometry" duality? Good opposite of a category of coalgebras?

So the category of affine schemes is dual to the category of commutative rings, Stone spaces are dual to Boolean algebras, localizable measurable spaces are dual to commutative Von Neumann algebras, and I'm sure there are many more examples. In general, a category of algebraic structures is going to be dual to some related category of geometric structures.

My question is, then: is there an analogous story for coalgbraic things? If I take a category of coalgebras for some comonad and flip the arrows around, will I get something interesting? Are there any good examples of this over familiar comonads (say, the costate comonad)?

If $R$ is a commutative ring, then the category of commutative bialgebras (resp. Hopf algebras) over $R$ is dual to the category of affine monoid (resp. group) schemes over $R$. This may be seen as an extension of the duality between commutative algebras over $R$ and affine schemes over $R$. However, I don't know any geometric description of coalgebras over $R$.