The transversal twistor space of a foliation $$F$$ of an even codimension is the bundle $$Z(F)$$ of the complex structures of the fibers of the transversalbundle of $$F$$ . On $$Z(F)$$ there exists a foliation $$\hat F$$ by covering spaces of the leaves of $$F$$ , and any Bottconnection of $$F$$ produces an ordered pair $$(\ell _1 ,\ell _2 )$$ of transversal almost complex structures of $$\hat F$$ . The existence of a Bott connection which yields a structure $$\ell $$ 1 that is projectable to the space of leaves isequivalent to the fact that $$F$$ is a transversallyprojective foliation. A Bott connection which yields a projectablestructure $$\ell $$ 2 exists iff $$F$$ isa transversally projective foliation which satisfies a supplementarycohomological condition, and, in this case, $$\ell $$ 1is projectable as well. $$\ell $$ 2 is never integrable.The essential integrability condition of $$\ell $$ 1 isthe flatness of the transversal projective structure of $$F$$ .