  Timelike Thomsen surfaces are timelike minimal surfaces that are also affine
minimal. In this paper, we make use of both the Lorentz conformal coordinates
and the null coordinates, and their respective representation theorems of
timelike minimal surfaces, to obtain a complete global classification of these
surfaces and to characterize them using a geometric invariant called lightlike
curvatures. As a result, we reveal the relationship between timelike Thomsen
surfaces, and timelike minimal surfaces with planar curvature lines. As an
application, we give a deformation of null curves preserving the
pseudo-arclength parametrization and the constancy of the lightlike curvatures.
