Here are some examples of problems which I believe have a reasonable claim of being "natural" (at least, they can be stated concisely) which (when phrased as decision problems) do not appear to be known in P nor known NP-complete. So if you want a dichotomy theorem you either need to be more precise about what you mean by "natural" in a way that excludes examples like these, or you need to look for more precisely defined subclasses of the natural problems (e.g. MSO formulas with the second-order quantification restricted to be existential and outside everything else) for which there might be more hope of a dichotomy.
I've made this community wiki because I think it could be a large list.
- Are there evenly many perfect matchings? (Should be $\oplus P$-complete)
- Who wins at strings-and-coins? (Might be PSPACE-complete)
- What is its list coloring or list edge coloring number? (Might be $\Pi^p_2$-complete)
- Is it a unit disk graph? A unit distance graph? (Might be $\exists \mathbb{R}$-complete)