Polarized signatures of orbiting hot spots: special relativity impact and probe of spacetime curvature

[Abridged] Context. The Galactic Center supermassive black hole is well known to exhibit transient peaks of flux density on a daily basis across the spectrum. Recent infrared and millimeter observations have strengthened the case for the association between these flares and circular orbital motion in the vicinity of the event horizon. The strongly polarized synchrotron radiation associated with these events leads to specific observables called QU loops, that is, looping motion in the Stokes QU plane of linear polarization. Aims. We want to deepen the understanding of the QU loops associated with orbiting hot spots. We compute such loops in Minkowski and Schwarzschild spacetimes in order to determine which aspects of the observed patterns are due to special- or general-relativistic phenomena. Results. We show that QU loops in Minkowski spacetime at low or moderate inclination i < 45 deg share all qualitative features of Schwarzschild QU loops: there exist QU loops for all setups considered (including face-on view and vertical magnetic field), there may be one or two QU loops per orbital period for a vertical magnetic field configuration, there are always two QU loops in case of a toroidal magnetic field. We provide analytical formulas in Minkowski spacetime to explain the details of this behavior. Moreover, we analyze the flux variation of the hot spot and show that it is dictated either by the angular dependence of the radiative transfer coefficients, or by relativistic beaming. In the former case, this can lead to extreme flux ratios even at moderate inclination. Finally, we highlight the increasing mirror asymmetry of the Schwarzschild QU track with increasing inclination and show that this behavior is a specific Schwarzschild feature caused by light bending.