  Given an annotated graph class $\mathcal{C}$, the
$\mathcal{C}$-blind-treewidth of a graph $G$ is the smallest integer $k$ such
that $G$ has a tree-decomposition where, for every $t\in V(T)$, if the
annotated graph $(G,\beta(t))$ does not belong to $\mathcal{C}$, then
$\beta(t)$ has size at most $k$. In this paper we focus on the annotated class
$\mathcal{B}$ of ``globally bipartite'' subgraphs and the class~$\mathcal{P}$
of ``globally planar'' subgraphs together with the \oddminor\ relation. For
each of the two parameters, $\mathcal{B}$-blind-treewidth and
${(\mathcal{B}\cup\mathcal{P})}$-blind-treewidth, we prove an analogue of the
celebrated Grid Theorem under the \oddminor\ relation. As a consequence we
obtain FPT-approximation algorithms for both parameters. We then provide
FPT-algorithms for \textsc{Maximum Independent Set} on graphs of bounded
$\mathcal{B}$-blind-treewidth and \textsc{Maximum Cut} on graphs of bounded
${(\mathcal{B}\cup\mathcal{P})}$-blind-treewidth.
