The spin-parity of the eta pi 0 pi 0 system from the charge-exchange transition pi(-) p --> eta pi(0) pi(0)n is analyzed for mass values up to 1.9 GeV. Data used in this analysis come from the GAMS-4000 multiphoton spectrometer irradiated by a 100-GeV/c pi(-) beam of the SPS accelerator at CERN. Experimental mass spectra show distinct signals from the pseudoscalar mesons eta(1295) and iota/eta(1440) and from the axial-vector meson D/f(1)(1285). The values found for their masses and widths agree well with the world-average values. The cross sections for the production of the above mesons are measured, and the relative branching fractions for some of their decays are estimated. The decay eta(1295) --> eta pi(0) pi is found to be dominated by the a(0)(980)pi 0 channel. On the other hand, the decay iota/eta(1440) --> a(0)(980)pi(0) is dominated by the eta(pi(0) pi(0))(s) channel, whereas the a(0)(980)pi 0 contribution is small. This suggests that iota is an exotic meson with a significant component of valence gluons.