  We investigate the fine structure in the energy spectrum of bilayer graphene
in the presence of various stacking defaults, such as a translational or
rotational mismatch. This fine structure consists of four Dirac points that
move away from their original positions as a consequence of the mismatch and
eventually merge in various manners. The different types of merging are
described in terms of topological invariants (winding numbers) that determine
the Landau-level spectrum in the presence of a magnetic field as well as the
degeneracy of the levels. The Landau-level spectrum is, within a wide parameter
range, well described by a semiclassical treatment that makes use of
topological winding numbers. However, the latter need to be redefined at zero
energy in the high-magnetic-field limit as well as in the vicinity of saddle
points in the zero-field dispersion relation.
