The dependence of dijet production on the virtuality of the exchanged photon, Q2, has been studied by measuring dijet cross sections in the range 0 ≲ Q2 < 2000 GeV2 with the ZEUS detector at HERA using an integrated luminosity of 38.6 pb-1. Dijet cross sections were measured for jets with transverse energy ETjet > 7.5 and 6.5 GeV and pseudorapidities in the photon-proton centre-of-mass frame in the range -3 < jet < 0. The variable xγobs, a measure of the photon momentum entering the hard process, was used to enhance the sensitivity of the measurement to the photon structure. The Q2 dependence of the ratio of low- to high-xγobs events was measured. Next-to-leading-order QCD predictions were found to generally underestimate the low-xγobs contribution relative to that at high xγobs. Monte Carlo models based on leading-logarithmic parton-showers, using a partonic structure for the photon which falls smoothly with increasing Q2, provide a qualitative description of the data. © Springer-Verlag / Società Italiana di Fisica 2004.