  We show that, under weak assumptions, the automorphism group of a ${\rm
CAT(0)}$ cube complex $X$ coincides with the automorphism group of Hagen's
contact graph $\mathcal{C}(X)$. The result holds, in particular, for universal
covers of Salvetti complexes, where it provides an analogue of Ivanov's theorem
on curve graphs of non-sporadic surfaces. This highlights a contrast between
contact graphs and Kim-Koberda extension graphs, which have much larger
automorphism group. We also study contact graphs associated to Davis complexes
of right-angled Coxeter groups. We show that these contact graphs are less
well-behaved and describe exactly when they have more automorphisms than the
universal cover of the Davis complex.
