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The scalar and tensor fluctuations produced during inflation can be correlated, if arising from the same underlying mechanism. In this paper we investigate such correlation in the model of axion inflation, where the rolling inflaton produces quanta of a U(1) gauge field which, in turn, source scalar and tensor fluctuations. We compute the primordial correlator of the curvature perturbation, ζ, with the amplitude of the gravitational waves squared, hijhij, at frequencies probed by gravitational wave detectors. This two-point function receives two contributions: one arising from the correlation of gravitational waves with the scalar perturbations generated by the standard mechanism of amplification of vacuum fluctuations, and the other coming from the correlation of gravitational waves with the scalar perturbations sourced by the gauge field. Our analysis shows that the latter effect is generally dominant. The correlator, normalized by the amplitude of ζ and of hijhij, turns out to be of the order of 10−2×(fequilNL)1/3, where fequilNL measures the scalar bispectrum sourced by the gauge modes.