Primitive positive constructions have been introduced in recent work of Barto, Opršal, and Pinsker to study the computational complexity of constraint satisfaction problems. Let P_fin be the poset which arises from ordering all finite relational structures by pp-constructability. This poset is infinite, but we do not know whether it is uncountable. In this article, we give a complete description of the restriction P_Boole of P_fin to relational structures on a two-element set. We use P_Boole to present the various complexity regimes of Boolean constraint satisfaction problems that were described by Allender, Bauland, Immerman, Schnoor and Vollmer.