  The magneto-oscillations in graphene bilayers are studied in the vicinity of
the K and K' points of the Brillouin zone within the four-band continuum model
ased on the simplest tight-binding approximation involving only the nearest
neighbor interactions. The model is employed to construct Landau plots for a
variety of carrier concentrations and bias strengths between the graphene
planes. The quantum-mechanical and quasiclassical approaches are compared. We
found that the quantum magneto-oscillations are only asymptotically periodic
and reach the frequencies predicted quasiclassically for high indices of Landau
levels. In unbiased bilayers the phase of oscillations is equal to the phase of
massive fermions. Anomalous behavior of oscillation phases was found in biased
bilayers with broken inversion symmetry. The oscillation frequencies again tend
to quasiclassically predicted ones, which are the same for $K$ and $K'$, but
the quantum approach yields the gate-tunable corrections to oscillation phases,
which differ in sign for K and K'. These valley-dependent phase corrections
give rise, instead of a single quasiclassical series of oscillations, to two
series with the same frequency but shifted in phase.
