Models of axion (monodromy) inflation are particularly interesting since they provide a natural justification for the flatness of the potential over a super-Planckian distance, namely the approximate shift-symmetry of the inflaton. In addition, most of the observational consequences are directly related to this symmetry and hence are correlated. Large tensor modes can be accompanied by the observable effects of a the shift-symmetric coupling $\phi F\tilde F$ to a gauge field. During inflation this coupling leads to a copious production of gauge quanta and consequently a very distinct modification of the primordial curvature perturbations. In this talk I compare these predictions with observations from power spectrum (WMAP7 + ACT), the bispectrum (WMAP7) and distortions to the CMB black body (COBE/FIRAS/TRIS) to obtain constraints on gauge production during inflation. Our findings suggest that our best bet for detecting evidence for a shift symmetric model of inflation is the (CMB) power spectrum. I will discuss additional (correlated) constraints on the axion model from periodic modulations to the power spectrum. I will briefly discuss the potential of position space correlation functions as a more direct measure of such modulations.