I'll put out the following possibly bold, possibly totally stupid conjectures. In any case, the statements below are not intended as mathematically rigorous statements, but may have some truth in them. Cheers!
Conjecture 1 Any naturally defined coherent sheaf on all smooth projective varieties is related to the sheaf of differentials via tensor operations and up to torsion.
A more concrete version is
Conjecture 2 Let $\mathscr L_X$ be a naturally defined line bundle on all smooth projective varieties $X$. Then some tensor power (possibly negative or zero) of $\mathscr L_X$ agrees with some tensor power of the canonical bundle.
Remark Jason pointed out that there are naturally defined torsion line bundles and Arend's idea of defining sheaves as push-forwards would produce sheaves supported on proper closed subvarieties. This is the main motivation for the "up to torsion" part of the first conjecture and for taking powers of the natural line bundle and the canonical bundle. Also notice that David Ben-Zvi's construction produces sheaves that satisfy these conjectures and the extensions in David Speyer's answer produce sheaves whose determinants are powers of the canonical sheaf.
To get to Conjecture 1 from Conjecture 2 one could argue the following way. Since the claim is "up to torsion" we can mod out by the torsion and assume that our sheaf is torsion-free. Now since we are on a smooth variety this implies that it is locally free in codimension $2$ and actually, again by the "up to torsion" principle, we may assume that it is reflexive, that is, take the reflexive hull, or in other words, the push-forward of the restriction to the open set where it is a locally free sheaf. In other words, we may perform all tensor operations as if we had locally free sheaves and in particular, the (reflexive hull of the) determinant will be a line bundle. In other words, up to torsion, we obtained a natural line bundle. If that is either the structure sheaf or the canonical bundle, then we're in business.
The reasoning I can offer for Conjecture 2 is the following: If there is a natural line bundle, then we can ask whether it is ample (or its inverse is) and for those varieties that it is we obtain a natural embedding (after taking some power). Once we have this we can look at the corresponding Hilbert schemes and try to construct moduli spaces. For those varieties on which this mysterious line bundle is not ample we can still define a corresponding Kodaira dimension and study Iitaka fibrations and eventually work toward a corresponding classification theory. I don't think any of this has happened except for the version using the canonical sheaf. I believe that suggests that there are no other non-trivial natural line bundles.