  The planar scattering amplitudes of $\mathcal{N} = 4$ super-Yang--Mills
theory display symmetries and structures which underlie their relatively simple
analytic properties such as having only logarithmic singularities and no poles
at infinity. Recent work shows in various nontrivial examples that the simple
analytic properties of the planar sector survive into the nonplanar sector, but
this has yet to be understood from underlying symmetries. Here we explicitly
show that for an infinite class of nonplanar integrals that covers all
subleading-color contributions to the two-loop four- and five-point amplitudes
of $\mathcal{N} = 4$ super-Yang--Mills theory, symmetries analogous to dual
conformal invariance exist. A natural conjecture is that this continues to all
amplitudes of the theory at any loop order.
