We study the structure of the space of diametrically complete sets in a finite dimensional normed space. In contrast to the Euclidean case, this space is in general not convex. We show that its starshapedness is equivalent to the completeness of the parallel bodies of complete sets, a property studied in Moreno and Schneider (Isr. J. Math. 2012, doi: 10.1007/s11856-012-0003-6), which is generically not satisfied. The space in question is, however, always contractible. Our main result states that in the case of a polyhedral norm, the space of translation classes of diametrically complete sets of given diameter is a finite polytopal complex. The proof makes use of a diagram technique, due to P. McMullen, for the representation of translation classes of polytopes with given normal vectors. The paper concludes with a study of the extreme diametrically complete sets.