It is well known that the maximal possible length of a minimal zero-sum sequence S in the group Z/nZ⊕Z/nZ equals 2n−1, and we investigate the structure of such sequences. We say that some integer n ≥ 2 has Property B, if every minimal zero-sum sequence S in Z/nZ ⊕ Z/nZ with length 2n − 1 contains some element with multiplicity n − 1. If some n ≥ 2 has Property B, then the structure of such sequences is completely determined. We conjecture that every n ≥ 2 has Property B, and we compare Property B with several other, already well-studied properties of zero-sum sequences in Z/nZ ⊕ Z/nZ. Among others, we show that if some integer n ≥ 6 has Property B, then 2n has Property B.