The complementarity between chiral perturbation theory and the linear sigma model is exploited to study π0π0 production in φ radiative decays, where the effects of the f0(980) scalar resonance, and those of its more controversial σ(500) partner, should become manifest via the φ → K+ K-(γ) → π0π0γ decay chain. The recently reported data on φ → π0π0γ coming from the VEPP-2M e+e- collider in Novosibirsk and the DAΦNE φ-factory in Frascati can be reasonably described in our approach, which we propose as a promising first step towards more detailed analyses. The f0(980) contribution, which appears as a moderately narrow peak at the high part of the dipion mass spectrum, can be interpreted as the isoscalar member of the scalar nonet with a large f0KK̄ coupling and an f0ππ coupling suppressed by almost ideal σ-f0 mixing. Indeed, the mixing angle in the flavor basis is found to be φs ≈ -6°, if the f0 propagator is approximated by a simple Breit-Wigner expression, or φs ≈ -9°, if an improved two-channel analysis is performed. The σ(500) resonance, which is then strongly coupled to pion pairs, yields a tiny contribution because, in our approach, its coupling to kaon pairs is proportional to mσ2 - mk2 and thus quite small.