  We study the double Gamow-Teller (DGT) strength distribution of $^{48}$Ca
with state-of-the-art large-scale nuclear shell model calculations. Our
analysis shows that the centroid energy of the DGT giant resonance depends
mostly on the isovector pairing interaction, while the resonance width is more
sensitive to isoscalar pairing. Pairing correlations are also key in
neutrinoless $\beta\beta$ ($0\nu\beta\beta$) decay. We find a simple relation
between the centroid energy and width of the $^{48}$Ca DGT giant resonance and
%the value of the $0\nu\beta\beta$ decay nuclear matrix element. More
generally, we observe a very good linear correlation between the DGT transition
to the ground state of the final nucleus and the $0\nu\beta\beta$ decay matrix
element. The correlation, which originates on the dominant short-range
character of both transitions, extends to heavier systems including several
$\beta\beta$ emitters, and also holds in energy-density functional results. Our
findings suggest that DGT experiments can be a very valuable tool to obtain
information on the value of $0\nu\beta\beta$ decay nuclear matrix elements.
