  For three-dimensional metals, Landau levels disperse as a function of the
magnetic field and the momentum wavenumber parallel to the field. In this
two-dimensional parameter space, it is shown that two conically-dispersing
Landau levels can touch at a diabolical point -- a Landau-Dirac point. The
conditions giving rise to Landau-Dirac points are shown to be magnetic
breakdown (field-induced quantum tunneling) and certain crystallographic
spacetime symmetry. Both conditions are realizable in topological nodal-line
metals, as we exemplify with CaP$_3$. A Landau-Dirac point reveals itself in
anomalous batman-like peaks in the magnetoresistance, as well as in the onset
of optical absorption linearly evolving to zero frequency as a function of the
field magnitude/orientation.
