  We present two new diagnostics based on the intrinsic shape alignments of
group/cluster size dark matter halos to disentangle the effect of $f(R)$
gravity from that of massive neutrinos. Using the snapshot data from a series
of the {\small DUSTGRAIN}-{pathfinder} $N$-body simulations for the Planck
$\Lambda$CDM cosmology and three $f(R)$ gravity models with massive neutrinos
($\nu$), we first determine the probability density functions of the alignment
angles between the shape orientations of massive halos and the minor principal
axes of the local tidal fields. The numerically obtained results turn out to
agree very well with the analytic formula derived under the assumption that the
anisotropic merging along the cosmic web induces the halo shape alignments. The
four cosmologies, which several standard diagnostics failed to discriminate,
are found to yield significantly different best-fit values of the single
parameter that characterizes the analytic formula. We also numerically
determine the spatial cross-correlations between the shape orientations of
neighbor group/cluster halos, and find them to be in good agreements with a
fitting formula characterized by two parameters, whose best-fit values are
found to substantially differ among the four models. We also discuss the
limitations and caveats of these new diagnostics that must be overcome for the
application to real observational data.
