In the description of resonant inelastic x-ray scattering (RIXS) from inversion-symmetric molecules the small core-level splitting is typically neglected. However, the spacing Δ between gerade and ungerade core levels in homonuclear diatomic molecules can be comparable with the lifetime broadening of the intermediate core-excited state Γ. We show that when Δ∼Γ the scattering becomes nonlocal in the sense that x-ray absorption at one atomic site is followed by emission at the other one. This is manifested in an unusual dependence of the RIXS cross section on the sum of the momenta of incoming and outgoing x-ray photons k+k′, contrary to the normal k-k′ dependence in the conventional local RIXS theory. The nonlocality of the scattering influences strongly the scattering angle and excitation energy dependence of the intensity ratio between parity forbidden and allowed RIXS channels. Numerical simulations for N2 show that this effect can readily be measured at present-day x-ray radiation facilities.