Convex bodies with pairs of sections associated by reflections

In this work we prove that if for a pair of convex bodies $K_1,K_2\subset {\mathbb{R}}^n$, $n\ge 3$, there exists a hyperplane $H$ and two distinct points $p_1$ and $p_2$ in ${\mathbb{R}}^n\setminus H$ such that for every $(n-2)$-plane $M\subset H$, there exists a reflection mapping the hypersection of $K_1$ defined by $\mathrm{a}\mathrm{f}\mathrm{f}\{p_1,M\}$ onto the hypersection of $K_2$ defined by $\mathrm{a}\mathrm{f}\mathrm{f}\{p_2,M\}$, then there exists a reflection which maps $K_1$ onto $K_2$.