We provide a geometric well-posedness theory for the Einstein equations within the class of weakly regular vacuum spacetimes with $T^2$-symmetry, as defined in the present paper, and we investigate their global causal structure. Our assumptions allow us to give a meaning to the Einstein equations under weak regularity as well as to solve the initial value problem under the assumed symmetry. First, introducing a frame adapted to the symmetry and identifying certain cancellation properties taking place...