  Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic
in Lorentzian conformal geometry which parallels the theory of Willmore
surfaces in $S^4$, are studied in this paper. We define two kinds of transforms
for such a surface, which produce the so-called left/right polar surfaces and
the adjoint surfaces. These new surfaces are again conformal Willmore surfaces.
For them holds interesting duality theorem. As an application spacelike
Willmore 2-spheres are classified. Finally we construct a family of homogeneous
spacelike Willmore tori.
