In prior joint work with Lewis, we developed a theory of enriched set-valued P -partitions to construct a K -theoretic generalization of the Hopf algebra of peak quasisymmetric functions. Here, we situate this object in a diagram of six Hopf algebras, providing a shifted version of the diagram of K -theoretic combinatorial Hopf algebras studied by Lam and Pylyavskyy. This allows us to describe new K -theoretic analogues of the classical peak algebra. We also study the Hopf algebras generated by Ikeda and Naruse’s K -theoretic Schur P - and Q -functions, as well as their duals. Along the way, we derive several product, coproduct, and antipode formulas and outline a number of open problems and conjectures.