We study the homological algebra of bimodules over involutive associative algebras. We show that Braun’s definition of involutive Hochschild cohomology in terms of the complex of involution-preserving derivations is indeed computing a derived functor: the Z/2-invariants intersected with the center. We then introduce the corresponding involutive Hochschild homology theory and describe it as the derived functor of the pushout of Z/2-coinvariants and abelianization
