Let $(R, m)$ be a commutative Noetherian Gorenstein local ring and let $M$ be an $R$-module. Let $^\perp M$ and $M^\perp$ be respectively the left and the right orthogonal classes of $M$ defined by vanishing of the $Ext$ functor; namely, $M^\perp$ consists of those $R$-modules $X$ such that $Ext^1_R(M, X)=0$, and $^\perp M$ is defined dually. Does any body have particular information on the (homological) properties of $^\perp M \cap M^\perp$? Any reference to relevant papers or books is highly appreciated even in case of particular assumptions on $M$.

$\DeclareMathOperator\Ext{Ext}$Let $(R, m)$ be a commutative Noetherian Gorenstein local ring and let $M$ be an $R$-module. Let ${^\perp M}$ and $M^\perp$ be respectively the left and the right orthogonal classes of $M$ defined by vanishing of the $\Ext$ functor; namely, $M^\perp$ consists of those $R$-modules $X$ such that $\Ext^1_R(M, X)=0$, and ${^\perp M}$ is defined dually. Does anybody have particular information on the (homological) properties of ${^\perp M} \cap M^\perp$? Any reference to relevant papers or books is highly appreciated even in case of particular assumptions on $M$.