<p>We characterize two objects by universal property: the derived de Rham complex and Hochschild homology together with its Hochschild–Kostant–Rosenberg (HKR) filtration. This involves endowing these objects with extra structure, built on notions of “homotopy-coherent cochain complex” and “filtered circle action” that we study here. We use these universal properties to give a conceptual proof that the associated graded of the HKR filtration identifies with the derived de Rham complex, as well as to give a new construction of the filtrations on cyclic, negative cyclic, and periodic cyclic homology that relate these invariants to derived de Rham cohomology.</p>