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We investigate the interaction of a high-intensity (~5.1018 W/cm2) and short (~65 fs) laser pulse with thin foils (from 0.08 to 105 mum) in a regime of ultrahigh contrast (> 1010). This paper shows that for thicknesses less than about 10 mum, proton acceleration from both sides of the target presents quite symmetric features. Proton bunches emitted from each side show similar maximum energies and spatial characteristics. Moreover, we show that for ultrahigh-contrast pulses, the efficient acceleration mechanism is related to the Brunei effect and not to the ponderomotive force. Simulations performed with a 2-D particle-in-cell code are in close agreement with all experimental data.