What is the spirit of the cylindric paradigm, as opposed to that of the polyadic one?

We give a categorial definition separating cylindric-like algebras from polyadic-like ones. Viewing the neat reduct operator as a functor, we show that it does not have a right adjoint in the former case, but it is strongly invertible in the second case. Several new results on amalgamation, and non finite axiomatizability are presented for both paradigms. A hitherto categorial equivalence is also given between relation algebras with quasi-projections and Nemeti's directed cylindric algebras for any dimension.