Where do functions for which $\partial_x\partial_y \neq \partial_y \partial_x$ fit in the theory of distributions?

Question

Consider $f$ defined on $\mathbb R^2$ by $$ f(x,y) = \begin{cases} \frac{xy(x^2-y^2)}{x^2+y^2} & (x,y)\neq 0 \\ 0 & (x,y)= 0\end{cases}$$ It is known that $\partial_x \partial_y f \neq \partial_y \partial_x f $. However, the following is also true: note that $f\in L^1_{\text {loc}}$, and so defines a distribution $T_f$, and since any test function $\phi$ is smooth, $$ \langle \partial_x \partial_y T_f,\phi\rangle = \langle T_f,\partial_y \partial_x\phi\rangle = \langle T_f,\partial_x \partial_y\phi\rangle = \langle \partial_y \partial_x T_f,\phi\rangle $$ So the distributional derivatives commute (and distributional derivatives always do). What's going on? I would guess that $T_{\partial_x \partial_y f}$, $T_{\partial_y \partial_x f}$, and $\partial_y\partial_x T_f$ are three distinct distributions. Is $\partial_y\partial_x T_f$ not given by a function?