Above, in a comment, Hao Sun asked if it was true that $\mathcal{H}om_X(\omega, \omega) = \mathcal{O}_X$. Here I will assume that $X$ is reduced and irreducible and $\omega$ is the canonical sheaf as discussed in S'andor's answer.
This second question is true if and only if the variety in question is S2 (so in particular, it holds in the Cohen-Macaulay case).
In fact, $\mathcal{H}om_X(\omega, \omega)$ is the S2-ification of $\mathcal{O}_X$ (and thus, Spec of it is the S2-ification of $X$).
EDIT: A reference for this last fact is Aoyama, "Some basic results on canonical modules", also see Aoyama, "On the depth and projective dimension of the canonical module".