  The low-temperature charge-density-wave (CDW) state in the layered organic
metals $\alpha $-(BEDT-TTF)$_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 alpha-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
alpha-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
magnetic-breakdown origin of the alpha-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.
