  We demonstrate that the Peccei-Quinn-electromagnetic anomaly coefficient
$\mathcal A$ can be directly measured from axion string-induced cosmic
birefringence by applying scattering transform to the anisotropic polarization
rotation of the cosmic microwave background. This breaks the degeneracy between
$\mathcal A$ and the effective number of string loops in traditional inference
analyses that are solely based on the spatial power spectrum of polarization
rotation. Carrying out likelihood-based parameter inference on mock rotation
realizations generated according to phenomenological string network models, we
show that scattering transform is able to extract enough non-Gaussian
information to clearly distinguish a number of discrete $\mathcal A$ values,
for instance $\mathcal{A}=1/9,\,1/3,\,2/3$, in the ideal case of noise-free
rotation reconstruction, and, to a lesser but interesting degree, at
reconstruction noise levels comparable to that expected for the proposed CMB-HD
concept. In the event of a statistical detection of cosmic birefringence by
Stage III or IV CMB experiments, our technique can be applied to test the
stringy nature of the birefringence pattern and extract fundamental information
about the smallest unit of charge in theories beyond the Standard Model.
