  We show that dual conformal symmetry, mainly studied in planar $\mathcal N =
4$ super-Yang-Mills theory, has interesting consequences for Feynman integrals
in nonsupersymmetric theories such as QCD, including the nonplanar sector. A
simple observation is that dual conformal transformations preserve unitarity
cut conditions for any planar integrals, including those without dual conformal
symmetry. Such transformations generate differential equations without raised
propagator powers, often with the right hand side of the system proportional to
the dimensional regularization parameter $\epsilon$. A nontrivial subgroup of
dual conformal transformations, which leaves all external momenta invariant,
generates integration-by-parts relations without raised propagator powers,
reproducing, in a simpler form, previous results from computational algebraic
geometry for several examples with up to two loops and five legs. By opening up
the two-loop three- and four-point nonplanar diagrams into planar ones, we find
a nonplanar analog of dual conformal symmetry. As for the planar case this is
used to generate integration-by-parts relations and differential equations.
This implies that the symmetry is tied to the analytic properties of the
nonplanar sector of the two-loop four-point amplitude of $\mathcal N = 4$
super-Yang-Mills theory.
