We have studied the phi -> a(0)(980)gamma process with the KLOE detector at the Frascati phi-factory DA Phi NE by detecting the phi -> eta pi(0)gamma decays in the final states with eta -> gamma gamma and eta -> pi(+)pi(-)pi(0). We have measured the branching ratios for both final states: Br(phi -> eta pi(0)gamma) = (7.01 +/- 0.10 +/- 0.20) x 10(-5) and (7.12 +/- 0.13 +/- 0.22) x 10(-5), respectively. We have also extracted the a(0)(980) mass and its couplings to eta pi(0), K+K-, and to the phi meson from the fit of the eta pi(0) invariant mass distributions using different phenomenological models.