I always had the impression that the principal bending axes of an isotropic square section must be the axes of symmetry. It turns out that any sectional axis can be the principal bending axes. For example, for the attached figure, the bending stiffness remain the same for these three sections.
Even for anisotropic and/or heterogeneous beam sections with general geometric shapes, as long as there are two equal principal bending compliance along a certain axis, then any axis can be the principal bending axis. See the interesting discussions at https://www.linkedin.com/feed/update/urn:li:activity:7222664865943801856/