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).
I am unaware of work specifically on convex polygon partitions, but perhaps if you
can specialize the algorithm in this paper to that simpler situation, its time complexity will
improve.
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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).
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