  Let $G$ be an additive abelian group. A sequence $S=g_1\cdot\ldots\cdot
g_{\ell}$ of terms from $G$ is a plus-minus weighted zero-sum sequence if there
are $\varepsilon_1,\ldots,\varepsilon_{\ell}\in\{-1,1\}$ such that
$\varepsilon_1 g_1+\ldots+\varepsilon_{\ell} g_{\ell}=0$. We first characterize
(in terms of $G$) when the monoid $\mathcal{B}_{\pm}(G)$ of plus-minus weighted
zero-sum sequences is Mori resp. Krull resp. finitely generated. After that we
study the Isomorphism and the Characterization Problem for monoids of
plus-minus weighted zero-sum sequences.
