Geometric interpretation of mixed partial derivatives?

Inspired by Ted Shifrin’s comment, here’s an attempt at an intuitive viewpoint. I’m not sure how much this counts as a “geometric interpretation”.

Consider a tiny square $ABCD$ of side length $h$, with $AB$ along the $x$-axis and $AD$ along the $y$-axis.

D---C
|   | h
A---B
  h

Then $f_x(A)$ is approximately $\frac1h\big(f(B)-f(A)\big)$, and $f_x(D)$ is approximately $\frac1h\big(f(C)-f(D)\big)$. So, assuming by $f_{xy}$ we mean $\frac\partial{\partial y}\frac\partial{\partial x}f$, we have
$$f_{xy}\approx\frac1h\big(f_x(D)-f_x(A)\big)\approx\frac1{h^2}\Big(\big(f(C)-f(D)\big)-\big(f(B)-f(A)\big)\Big).$$
Similarly,
$$f_{yx}\approx\frac1{h^2}\Big(\big(f(C)-f(B)\big)-\big(f(D)-f(A)\big)\Big).$$
But those two things are the same: they both correspond to the “stencil”
$$\frac1{h^2}\begin{bmatrix} -1 & +1\\ +1 & -1 \end{bmatrix}.$$