We study transformation properties of momentum space renormalization group
transformations. The renormalization group transformations are composed of
an integral over a high momentum field with a scale transformation of a low
momentum field. As an example, we consider perturbations of a massless rea
l euclidean free field phi with a smooth momentum space regulator. We show
that this renormalization group admits two equivalent formulations called t
he massless picture and the massive picture. In the massive picture, the re
normalization group is shown to have a symmetry. The symmetry consists of g
lobal (space-time independent) scale transformations of the field composed
with a certain (gaussian) integral. We then translate the symmetry back to
the massless picture. The relation between the symmetry and the notion of a
n anomalous dimension is briefly discussed.