We report on the magnetic-field-induced variations of the microwave surface resistance,Rs, ina polycrystalline MgB2 sample, at different values of temperature. We have detected a magnetic hysteresis inRs, which exhibits an unexpected plateau on decreasing the DC magneticfield below a certain value. In particular, at temperatures nearTc the hysteresis manifests itself only through the presence of the plateau.Although we do not quantitatively justify the anomalous shape of the magnetichysteresis, we show that the results obtained in the reversible region of theRs(H) curve can be quite well accounted for by supposing that, in this range of magnetic field, theπ gap is almost suppressed by the applied field and, consequently, all theπ-band charge carriers are quasiparticles. On this hypothesis, we have calculatedRs(H) supposing that fluxons assume a conventional (single core) structure and theflux dynamics can be described in the framework of conventional models.From the fitting of the experimental results, we determine the values ofHc2π(T) at temperaturesnear Tc. In our opinion, the most important result of our investigation is that, at least at temperatures nearTc, the value of the applied field that separates the reversible and irreversible regions of theRs(H) curvesis just Hc2π(T); a qualitative discussion of the possible reasons for this finding is given.