Let $n \in \mathbb{N}$, let $M$ be a smooth $n$-dimensional manifold, let $m \in \mathbb{R}$, and let $A_h \in \Psi_h^m(M;\Omega^{1/2})$ be a properly supported semiclassical pseudodifferential operator of order $m$ acting on half-densities for $0<h\leq h_0$. Let $(U,\varphi)$ and $(V,\psi)$ be coordinate charts with $U \cap V \neq \varnothing$, set $X:=\varphi(U\cap V)\subset \mathbb{R}^n$ and $Y:=\psi(U\cap V)\subset \mathbb{R}^n$, write $\kappa := \psi \circ \varphi^{-1}: X \to Y$ for the coordinate change, and let $\theta := \kappa^{-1}:Y\to X$. Assume that, after the standard half-density trivializations in the two charts, the Schwartz kernel of $A_h$ is represented near the diagonal in $X \times X$ by a local left quantization, understood as a semiclassical oscillatory integral, with full symbol $a: X \times \mathbb{R}^n \times (0,h_0] \to \mathbb{C}$, $a \in S^m(X\times\mathbb{R}^n)$, modulo an $O(h^\infty)$ smoothing remainder, and near the diagonal in $Y \times Y$ by a local left quantization, also understood as a semiclassical oscillatory integral, with full symbol $b: Y \times \mathbb{R}^n \times (0,h_0] \to \mathbb{C}$, $b \in S^m(Y\times\mathbb{R}^n)$, modulo an $O(h^\infty)$ smoothing remainder. Let $a_0$ and $b_0$ denote the classes of $a$ and $b$ modulo $hS^{m-1}$ in these two coordinate systems. Then, for every $x \in X$, every $y=\kappa(x)$, and every $\eta \in \mathbb{R}^n$, under the standard coordinate identification of cotangent fibres, \begin{align*} b_0(y,\eta)=a_0\bigl(x,(d\kappa_x)^*\eta\bigr). \end{align*} Equivalently, the principal symbol of $A_h$ defines a coordinate-independent class \begin{align*} \sigma_h^m(A_h) \in S^m(T^*M)/hS^{m-1}(T^*M). \end{align*} For operators acting on functions, the same coordinate-invariance statement holds after a density convention has been fixed; lower-order terms of the full symbol may depend on that convention.