  We investigate the distortion of Assouad dimension and the Assouad spectrum
under Euclidean quasiconformal maps. Our results complement existing
conclusions for Hausdorff and box-counting dimension due to
Gehring--V\"ais\"al\"a and others. As an application, we classify polynomial
spirals $S_a:=\{x^{-a}e^{\mathbf{i} x}:x>0\}$ up to quasiconformal equivalence,
up to the level of the dilatation. Specifically, for $a>b>0$ we show that there
exists a quasiconformal map $f$ of $\mathbb{C}$ with dilatation $K_f$ and
$f(S_a)=S_b$ if and only if $K_f \ge \tfrac{a}{b}$.
