  The S-matrix of a theory often exhibits symmetries which are not manifest
from the viewpoint of its Lagrangian. For instance, powerful constraints on
scattering amplitudes are imposed by the dual conformal symmetry of planar 4d
$\mathcal{N}=4$ super Yang-Mills theory and the ABJM theory. Motivated by this,
we investigate the consequences of dual conformal symmetry in six dimensions,
which may provide useful insight into the worldvolume theory of M5-branes (if
it enjoys such a symmetry). We find that 6d dual conformal symmetry uniquely
fixes the integrand of the one-loop 4-point amplitude, and its structure
suggests a Lagrangian with more than two derivatives. On integrating out the
loop momentum in $6-2 \epsilon$ dimensions, the result is very similar to the
corresponding amplitude of $\mathcal{N}=4$ super Yang-Mills theory. We confirm
this result holographically by generalizing the Alday-Maldacena solution for a
minimal area string in Anti-de Sitter space to a minimal volume M2-brane ending
on pillow-shaped Wilson surface in the boundary whose seams correspond to a
null-polygonal Wilson loop. This involves careful treatment of a prefactor
which diverges as $1/\epsilon$, and we comment on its possible interpretation.
We also study 2-loop 4-point integrands with 6d dual conformal symmetry and
speculate on the existence of an all-loop formula for the 4-point amplitude.
