  When $\mathcal{T}$ is a tree decomposition of a graph $G$, we write
$\mu(\mathcal{T})$ for the maximum size of an induced matching in $G$ all of
whose edges intersect one bag of $\mathcal{T}$. The induced matching treewidth
of a graph $G$ is the minimum value of $\mu(\mathcal{T})$ over all tree
decompositions $\mathcal{T}$ of~$G$. Classes of graphs with bounded induced
matching treewidth admit polynomial-time algorithms for a number of problems,
including INDEPENDENT SET, $k$-COLORING, ODD CYCLE TRANSVERSAL, and FEEDBACK
VERTEX SET. In this paper, we focus on structural properties of such classes.
  First, we show that graphs with bounded induced matching treewidth that
exclude a fixed biclique as an induced subgraph have bounded tree-independence
number, which is another well-studied parameter defined in terms of tree
decompositions. This sufficient condition about excluding a biclique is also
necessary, as bicliques have unbounded tree-independence number. Second, we
show that graphs with bounded induced matching treewidth that exclude a fixed
clique have bounded chromatic number. That is, classes of graphs with bounded
induced matching treewidth are $\chi$-bounded. Our results confirm two
conjectures from a recent manuscript of Lima et al. [arXiv 2024].
