  For a finite abelian group $(G,+)$ the Harborth constant is defined as the
smallest integer $\ell$ such that each squarefree sequence over $G$ of length
$\ell$ has a subsequence of length equal to the exponent of $G$ whose terms sum
to $0$. The plus-minus weighted Harborth constant is defined in the same way
except that the existence of a plus-minus weighted subsum equaling $0$ is
required, that is, when forming the sum one can chose a sign for each term. The
inverse problem associated to these constants is the problem of determining the
structure of squarefree sequences of maximal length that do not yet have such a
zero-subsum. We solve the inverse problems associated to these constant for
certain groups, in particular for groups that are the direct sum of a cyclic
group and a group of order two. Moreover, we obtain some results for the
plus-minus weighted Erd\H{o}s--Ginzburg--Ziv constant.
