  Let $(G,+)$ be a finite abelian group. Then, $\so(G)$ and $\eta(G)$ denote
the smallest integer $\ell$ such that each sequence over $G$ of length at least
$\ell$ has a subsequence whose terms sum to $0$ and whose length is equal to
and at most, resp., the exponent of the group. For groups of rank two, we study
the inverse problems associated to these constants, i.e., we investigate the
structure of sequences of length $\so(G)-1$ and $\eta(G)-1$ that do not have
such a subsequence. On the one hand, we show that the structure of these
sequences is in general richer than expected. On the other hand, assuming a
well-supported conjecture on this problem for groups of the form $C_m \oplus
C_m$, we give a complete characterization of all these sequences for general
finite abelian groups of rank two. In combination with partial results towards
this conjecture, we get unconditional characterizations in special cases.
