  In an application of the notion of twisting structures introduced by Hess and
Lack, we define twisted composition products of symmetric sequences of chain
complexes that are degreewise projective and finitely generated. Let Q be a
cooperad and let BP be the bar construction on the operad P. To each morphism
of cooperads g from Q to BP is associated a P-co-ring, K(g), which generalizes
the two-sided Koszul and bar constructions. When the co-unit from K(g) to P is
a quasi-isomorphism, we show that the Kleisli category for K(g) is isomorphic
to the category of P-algebras and of their morphisms up to strong homotopy, and
we give the classifying morphisms for both strict and homotopy P-algebras.
Parametrized morphisms of (co)associative chain (co)algebras up to strong
homotopy are also introduced and studied, and a general existence theorem is
proved. In the appendix, we study the particular case of the two-sided Koszul
resolution of the associative operad.
