[This was first posted on MSE but did not get any answer. I apologize if it is not suited for MO.]
Let $C$ be a curve, i.e. a purely one-dimensional scheme, embedded in a smooth projective threefold $X$. For a coherent sheaf $E$ of codimension $c$ on $X$, let $E^D=\mathscr Ext_X^c(E,\omega_X)$ be the Grothendieck dual of $E$. It is a reflexive sheaf of codimension $c$ on $X$. Reflexive means that the natural map to the double dual is an isomorphism. Let $\mathscr O_C$ be the structure sheaf of $C$, viewed as a torsion sheaf on $X$.
Question. For which curves $C\subset X$ is $\mathscr O_C$ reflexive?
Certainly, when $C$ is smooth. Let us assume $C$ is singular. I was thinking that maybe if $C$ is a local complete intersection then the first dual $\mathscr O_C^D$ could already be isomorphic to $\mathscr O_C$. Is this true? If so, then in particular dualizing twice does not do anything. But probably someone knows a more convincing (or even more true) statement. Thank you for any help!