The low-temperature charge-density-wave (CDW) state in the layered organic metals α-(BEDT-TTF) 2 MHg(SCN) 4 has been studied by means of the Shubnikov -de Haas and de Haas -van Alphen effects. In addition to the dominant α-frequency, which is also observed in the normal state, both the magnetoresistance and magnetic torque possess a slowly oscillating component. These slow oscillations provide a firm evidence for the CDW-induced reconstruction of the original cylindrical Fermi surface. The α-oscillations of the interlayer magnetoresistance exhibit an anomalous phase inversion in the CDW state, whereas the de Haas -van Alphen signal maintains the normal phase. We argue that the anomaly may be attributed to the magneticbreakdown origin of the α-oscillations in the CDW state. A theoretical model illustrating the possibility of a phase inversion in the oscillating interlayer conductivity in the presence of a spatially fluctuating magnetic breakdown gap is proposed.