In this lecture, we mainly address the conditions for reconstructing a convex polygon with parallel opposite sides from its so-called defects of symmetry. This concept translates properties of ovals, defined by pairs of parallel tangents, into discrete polygon space. We discuss the discrete equivalents of cusps, Wigner caustic, Centre Symmetry Set, and affine lambda-equidistant sets. This is a joint work with B. Murawski, T. Wieczorek, and M. Zwierzynski.