  For any given dimension $d$, all reflexive $d$-polytopes can be found (in
principle) as subpolytopes of a number of maximal polyhedra that are defined in
terms of $(d+1)$-tuples of integers (weights), or combinations of $k$-tuples of
weights with $k<d+1$. We present the results of a complete classification of
sextuples of weights pertaining to the construction of all reflexive polytopes
in five dimensions. We find 322 383 760 930 such weight systems. 185 269 499
015 of them give rise directly to reflexive polytopes and thereby to mirror
pairs of Calabi-Yau fourfolds. These lead to 532 600 483 distinct sets of Hodge
numbers.
