There exist convex quadrilaterals which have no such splitting. And there is an $O(n^3)$ algorithm to decide if such a splitting exists for a (nonconvex) $n$-gon. See the paper by Dania El-Khechen, Thomas Fevens, John Iacono, and Günter Rote, "Partitioning a polygon into two mirror congruent pieces." In Proc. 20th Canad. Conf. Comput. Geom., pages 131-134, August 2008 (PDF download link).
There exist convex quadrilaterals which have no such splitting. And there is an $O(n^3)$ algorithm to decide if such a splitting exists for a (nonconvex) $n$-gon. See the paper by Dania El-Khechen, Thomas Fevens, John Iacono, and Günter Rote, "Partitioning a polygon into two mirror congruent pieces." In Proc. 20th Canad. Conf. Comput. Geom., pages 131-134, August 2008 (PDF download link).