Let $X$ be a Noetherian scheme that is not Gorenstein but possesses a dualizing complex of coherent sheaves. Then (if I understand these matters and the answer to the question Characterization of schemes whose dualizing complex is perfect correctly) the bounded derived category $D^b_{coh}(X)$ of coherent sheaves on $X$ is self-dual, but this duality does not send (all) perfect complexes into perfect ones (recall that an object of $D^b_{coh}(X)$ is a perfect complex if it is locally quasi-isomorphic to a bounded complex of free sheaves).

My question is: did anybody study the image of the triangulated subcategory of perfect complexes under coherent duality? Is there any name for this subcategory of $D^b_{coh}(X)$?