Generally, the differential operator $d$ takes differential $k$-forms to $(k+1)$-forms. In $\mathbb{R}^n$ (or more generally on an oriented Riemannian manifold), we can identify vector fields with $1$-forms and $(n-1)$-forms, and functions with $0$-forms and $n$-forms. If we write $\phi$ for the map between vector fields and 1-forms, and $\psi$ for the map between vector fields and $(n-1)$-forms, then the composite
function $=$ 0-form $\xrightarrow{d}$ 1-form $\xrightarrow{\phi}$ vector field
is the gradient, and the composite
vector field $\xrightarrow{\psi}$ $(n-1)$-form $\xrightarrow{d}$ $n$-form $=$ function
is the divergence. When $n=3$, we also have the composite
vector field $\xrightarrow{\phi}$ 1-form $\xrightarrow{d}$ 2-form $\xrightarrow{\psi}$ vector field
which is the curl. And when $n=2$, we have the composite
vector field $\xrightarrow{\phi}$ 1-form $\xrightarrow{d}$ 2-form $=$ function
which is the "scalar curl" that appears in Green's theorem. Note that it differs from the $n=2$ case of the divergence only in that it involves $\phi$ rather than $\psi$. When $n=2$ both $\phi$ and $\psi$ relate vector fields to 1-forms, but they do it differently; we have $\phi(\langle F,G\rangle) = F\,dx + G\,dy$, while $\psi(\langle F,G\rangle) = -G\,dx + F\,dy$.
This suggests that there is another way to put these together when $n=2$:
function $=$ 0-form $\xrightarrow{d}$ 1-form $\xrightarrow{\psi}$ vector field
This looks like the $n=2$ case of the gradient, but involves $\psi$ rather than $\phi$. In coordinates, it takes a function $f$ to the vector field $\langle \frac{\partial f}{\partial y} , -\frac{\partial f}{\partial x}\rangle$.
I've never seen this operation before; does it have a name? It seems potentially interesting, e.g. when we specialize the generalized Stokes' theorem to it we get an FTC for its flux line integrals along curves, analogous to the usual FTC for the flow line integral of a gradient field.