  We study the second order invariants of a Lorentzian surface in
$\mathbb{R}^{2,2},$ and the curvature hyperbolas associated to its second
fundamental form. Besides the four natural invariants, new invariants appear in
some degenerate situations. We then introduce the Gauss map of a Lorentzian
surface and give an extrinsic proof of the vanishing of the total Gauss and
normal curvatures of a compact Lorentzian surface. The Gauss map and the second
order invariants are then used to study the asymptotic directions of a
Lorentzian surface and discuss their causal character. We also consider the
relation of the asymptotic lines with the mean directionally curved lines. We
finally introduce and describe the quasi-umbilic surfaces, and the surfaces
whose four classical invariants vanish identically.
