The combustion method was used to prepare a precursor powder of an iron-gallium oxide compound which was further heat-treated in order to obtain a set of Fe1+xGa2-xO4 nanoparticles. All samples have a cubic spinel-type structure (space group Fd3[combining macron]m) and the particle size varies from 1.8 to 28.0 nm depending on the treatment conditions. From the comparative analysis by XRD, EDS, and Raman and Mössbauer spectroscopy the creation of a new spinel phase γ-FeGaO3, which was mainly located on the particle surface, was established. As a result, the composition consists of a FeGa2O4 core covered by a FeGaO3 shell. The relative content of FeGa2O4/FeGaO3 compounds in the composites can be varied by heat treatment. The maximum in the ZFC magnetization curves appeared in all samples at about 20-30 K corresponding to the spin-freezing temperature Tsg, which is much higher than in the bulk compound with a pure inverse spinel structure (Ga)[FeGa]O4. The values of effective Curie temperature ΘC for the Fe1+xGa2-xO4 nanoparticles are rather high and positive, indicating a ferromagnetic interaction between iron ions. The high values of the magnetic frustration parameter f = ΘC/Tsg (up to 7) indicate a high degree of magnetic frustration. The low temperature Mössbauer data reveal the magnetic ordering of Fe ions in all samples with the magnetic transition at about 20-26 K depending on the particle size. The specific features of the Mössbauer parameters indicate the properties of non-homogeneous magnetic systems with frustrated interactions specific to spin-glasses. The magnetic system behaves as a spin-glass below Tsg and it is superparamagnetic above Tsg. Such a system is called a "super-spin-glass". The anisotropy energy Eanis strongly depends on the content of Fe(2+) and Fe(3+) ions which contribute to the magnetocrystalline Ecryst and exchange Eex anisotropies, respectively. The anisotropy energy can be tuned by variation of the content of the (FeGaO3)-(FeGa2O4) phases in these complex composites.