Folded symplectic forms in contact topology

We establish the relationship between folded symplectic forms and convex hypersurface theory in contact topology. As an application, we use convex hypersurface theory to reprove and strengthen the existence result for folded symplectic forms due to Cannas da Silva, and we generalize to all even dimensions Baykur's $4$-dimensional existence result of folded Weinstein structures and folded Lefschetz fibrations.