  Adinkras are combinatorial objects developed to study supersymmetry
representations. Gates et al. introduced the "gadget" as a function of pairs of
adinkras, obtaining some mysterious results for $(n=4, k=1)$ adinkras with
computer-aided computation. Specifically, very few values of the gadget
actually appear, suggesting a great deal of symmetry in these objects. In this
paper, we compute gadgets symbolically and explain some of these observed
phenomena with group theory and combinatorics. Guided by this work, we give
some suggestions for generalizations of the gadget to other values of the $n$
and $k$ parameters.
