  We show that Willwacher's cyclic formality theorem can be extended to
preserve natural Gravity operations on cyclic multivector fields and cyclic
multidifferential operators. We express this in terms of a homotopy Gravity
quasi-isomorphism with explicit local formulas. For this, we develop operadic
tools related to mixed complexes and cyclic homology and prove that the operad
$\mathsf M_\circlearrowleft$ of natural operations on cyclic operators is
formal and hence quasi-isomorphic to the Gravity operad.
