  We study double gamma ($\gamma\gamma$) decay nuclear matrix elements (NMEs)
for a wide range of nuclei from titanium to xenon, and explore their relation
to neutrinoless double-beta ($0\nu\beta\beta$) NMEs. To favor the comparison,
we focus on double-magnetic dipole transitions in the final $\beta\beta$
nuclei, in particular the $\gamma\gamma$ decay of the double isobaric analog of
the initial $\beta\beta$ state into the ground state. For the decay with
equal-energy photons, our large-scale nuclear shell model results show a good
linear correlation between the $\gamma\gamma$ and $0\nu\beta\beta$ NMEs. Our
analysis reveals that the correlation holds for $\gamma\gamma$ transitions
driven by the spin or orbital angular momentum due to the dominance of
zero-coupled nucleon pairs, a feature common to $0\nu\beta\beta$ decay. Our
shell-model findings point out the potential of future $\gamma\gamma$ decay
measurements to constrain $0\nu\beta\beta$ NMEs, which are key to answer
fundamental physics questions based on $0\nu\beta\beta$ experiments.
