  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 most probable
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
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.
