translation invariance disguises itself as an “accidental” symmetry. As a consequence, the symmetry multiplets — i.e. the Landau levels — are infinitely degenerate. In order to further investigate the nature of the accidental symmetry, in [@Dul66] the charged particle in the magnetic field was coupled to the origin by an $r^2$ harmonic oscillator potential. This explicitly breaks translation invariance and thus reduces the degeneracy to a finite amount, while rotation invariance remains intact. In this paper, we do the opposite, i.e. we explicitly break rotation invariance, while leaving translation invariance (and hence the accidental symmetry) intact by putting the system on a torus. Interestingly, the Polyakov loops, which are a consequence of the non-trivial holonomies of the torus, give rise to non-trivial Aharonov-Bohm phases which are observable at the quantum but not at the classical level. Analogous to the quantum mechanical breaking of CP invariance due to the $\theta$-vacuum angle in non-Abelian gauge theories, here two self-adjoint extension parameters $\theta_x$ and $\theta_y$ explicitly break the continuous translation invariance of the classical problem down to a discrete magnetic translation group [@Zak64]. This reduces the degeneracy to a finite amount, and allows us to further investigate the nature of the accidental symmetry. 