  We survey the role of symmetry in diffeomorphic registration of landmarks,
curves, surfaces, images and higher-order data. The infinite dimensional
problem of finding correspondences between objects can for a range of concrete
data types be reduced resulting in compact representations of shape and spatial
structure. This reduction is possible because the available data is incomplete
in encoding the full deformation model. Using reduction by symmetry, we
describe the reduced models in a common theoretical framework that draws on
links between the registration problem and geometric mechanics. Symmetry also
arises in reduction to the Lie algebra using particle relabeling symmetry
allowing the equations of motion to be written purely in terms of Eulerian
velocity field. Reduction by symmetry has recently been applied for
jet-matching and higher-order discrete approximations of the image matching
problem. We outline these constructions and further cases where reduction by
symmetry promises new approaches to registration of complex data types.
