It is well known that a distribution supported on $\{0\}\in \mathbb{R}^2$ is a finite linear combination of derivatives of the unit measure $\delta_0$. Moreover, if $u \in \mathcal{D}'(\mathbb{R}^2)$ is of finite order $k$ with $\text{supp}(u) \subset \mathbb{R} \times \{0\}$, then it is of the form \begin{align*} u = \sum_{j \leq k} u_j \otimes \partial^j \delta_0, \end{align*} where $u_j \in \mathcal{D}'(\mathbb{R})$ are distributions of finite order $k-j$. But what if instead the support of $u$ is a cross? That is, for example, $\text{supp}(u) = ([-1, 1] \times \{0\}) \cup (\{0\} \times [-1, 1])$.
It is not clear to me if there is (or should be) a similar decomposition. If $u$ is supported on a cross, then $0$ can be cut out, leaving a sum of two distributions that can be represented as above, plus a remainder supported in an arbitrarily small neighborhood of $0$. But it does not seem possible to just get rid of this term by passing to a limit. I guess I am asking if in this case \begin{align*} u = \sum_{j \leq k} u_j^{(1)} \otimes \partial^j \delta_0 + \sum_{j \leq k} \partial^j \delta_0 \otimes u^{(2)}_j, \end{align*} where $u_j^{(1)}$ and $u_j^{(2)}$ are suitable distributions of finite order $k-j$ supported in $[-1,1]$. Are there any counterexamples to this?