Let $X$ be a Noetherian scheme that is not Gorenstein but possesses a dualizing complex $D$ 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 $D_X$ 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 $D^{perf}(X)\subset D^b_{coh}(X)$ under the coherent duality $D_X$? Is there any name for this subcategory $D_X(D^{perf}(X))\subset D^b_{coh}(X)$? Is it true that $D_X(D^{perf}(X))\cong D^{perf}(X)\otimes D$? Does it follow that $D_X$ induces an equivalence $D^{perf}(X)^{op}\cong D^{perf}(X)$?