  We show that the cylindrical symmetry of the eigenvectors of the photon
position operator with commuting components, x, reflects the E(2) symmetry of
the photon little group. The eigenvectors of x form a basis of localized states
that have definite angular momentum, J, parallel to their common axis of
symmetry. This basis is well suited to the description of "twisted light" that
has been the subject of many recent experiments and calculations. Rotation of
the axis of symmetry of this basis results in the observed Berry phase
displacement. We prove that {x1,x2,J3} is a realization of the two dimensional
Euclidean e(2) algebra that effects genuine infinitesimal displacements in
configuration space.
