In this study we address the role of surface anisotropy on the hysteretic properties of magnetite Fe3O4 nanoparticles and the circumstances yielding both horizontal and vertical shifts in the hysteresis loops. Our analysis involves temperature dependence and particle size effects. Different particle sizes ranging from 2 up to 7 nm were considered. Our theoretical framework is based on a three-dimensional classical Heisenberg model with nearest magnetic neighbor interactions involving tetrahedral (A) and octahedral (B) irons. Cubic magnetocrystalline anisotropy for core spins, single-ion site anisotropy for surface spins, and interaction with a uniform external magnetic field were considered. Our results revealed the onset of low temperature exchange bias field, which can be positive or negative at high enough values of the surface anisotropy constant (KS). Susceptibility data, computed separately for the core and the surface, suggest differences in the hard-soft magnetic character at the core-surface interface. Such differences are KS-driven and depend on the system size. Such a hard-soft interplay, via the surface anisotropy, is the proposed mechanism for explaining the observed exchange bias phenomenology. Our results indicate also that the strongly pinned spins at high enough surface anisotropy values are responsible for both the horizontal and vertical shifts in the hysteresis loops. The dependences of the switching and exchange bias fields with the surface anisotropy and temperature are finally discussed.