What do dichotomy theorems feed on? It is well known that certain classes of NP-problems have dichotomy theorems, which guarantee that every task in the class is either NP-complete or is in P. The best known such result is 
Schaefer's dichotomy 
theorem, along with a number of generalizations.

My understanding is that proving these dichotomy theorems is not really easy. I wonder, if there is any relatively short explanation for why certain classes have dichotomy theorems, while others do not? What is the essential problem structure that makes these theorems possible? Or perhaps there is no such clearly understood structure, rather it is a mystery in each case why the class does or does not have a dichotomy theorem?
For the case of Schaefer's dichotomy theorem, informally, the universal expressive power of Boolean CNF formulas built from non-Schaefer logical relations is behind the dichotomy. Every logical relation is definable by such formula using existential quantifier. This is stated formally in Schaefer's expressibility theorem (Theorem 2.5).