  We establish a structural understanding of the involutive Heegaard Floer
homology for all linear combinations of almost-rational (AR) plumbed
three-manifolds. We use this to show that the Neumann-Siebenmann invariant is a
homology cobordism invariant for all linear combinations of AR plumbed homology
spheres. As a corollary, we prove that if $Y$ is a linear combination of AR
plumbed homology spheres with $\mu(Y) = 1$, then $Y$ is not torsion in the
homology cobordism group. A general computation of the involutive Heegaard
Floer correction terms for these spaces is also included.
