Let $n$ be a positive integer, let $U \subset \mathbb{R}^n$ be open, and let $m \in \mathbb{R}$. Fix a smooth cutoff function $\chi: \mathbb{R}^n \to [0,1]$ such that $\chi(\xi)=0$ for $|\xi| \leq 1/2$ and $\chi(\xi)=1$ for $|\xi| \geq 1$. Let $a,b: U \times \mathbb{R}^n \to \mathbb{C}$ be classical symbols in $S_{\mathrm{cl}}^m(U \times \mathbb{R}^n)$ with asymptotic expansions $a \sim \sum_{j=0}^{\infty} a_{m-j}$ and $b \sim \sum_{j=0}^{\infty} b_{m-j}$, meaning that for every $N \in \mathbb{N}$ the cutoff finite sums $\sum_{j=0}^{N-1}\chi(\xi)a_{m-j}(x,\xi)$ and $\sum_{j=0}^{N-1}\chi(\xi)b_{m-j}(x,\xi)$ define smooth symbols on $U \times \mathbb{R}^n$ and the corresponding remainders lie in $S^{m-N}(U \times \mathbb{R}^n)$. Here for every $j \in \mathbb{N} \cup \{0\}$ the functions $a_{m-j}, b_{m-j}: U \times (\mathbb{R}^n \setminus \{0\}) \to \mathbb{C}$ are smooth and homogeneous of degree $m-j$ in the covariable $\xi$. If $a_{m-j}=b_{m-j}$ on $U \times (\mathbb{R}^n \setminus \{0\})$ for every $j \in \mathbb{N} \cup \{0\}$, then $a-b \in S^{-\infty}(U \times \mathbb{R}^n)$. Conversely, if $a \in S_{\mathrm{cl}}^m(U \times \mathbb{R}^n)$ has homogeneous components $(a_{m-j})_{j \geq 0}$ and $r \in S^{-\infty}(U \times \mathbb{R}^n)$, then $a+r \in S_{\mathrm{cl}}^m(U \times \mathbb{R}^n)$ and has the same homogeneous components $(a_{m-j})_{j \geq 0}$ as $a$.