On bodies with congruent sections or projections

In this paper, we construct two convex bodies $K$ and $L$ in ${\mathbb{R}}^n$, $n\ge 3$, such that their projections $K\mathrm{\mid }H$, $L\mathrm{\mid }H$ onto every subspace $H$ are congruent, but nevertheless, $K$ and $L$ do not coincide up to a translation or a reflection in the origin. This gives a negative answer to an old conjecture posed by Nakajima and S"uss.