Single‐crystalline Fe1+xTe iron telluride with off‐stoichiometric iron has been synthesized by the Bridgman method. The X‐ray diffraction and wave‐length‐dispersive X‐ray electron‐probe microanalysis characterization have shown Fe1.125Te stoichiometry of the samples. Spin‐polarized ab initio calculations of the electric field gradients around interstitial iron atoms for Fe1.125Te have shown that in the first and second coordination rings around interstitial iron, the spin and electron densities are strongly perturbed against the stoichiometric ones. Together with the interstitial iron this gives rise to three kinds of iron centers making up a round‐corner plaquette. The room‐temperature Mössbauer spectra measured at different incidence angles of gamma‐radiation are satisfactorily fitted utilizing the hyperfine parameters, calculated within the plaquette model. The low‐temperature data are well described with the assumption of an incommensurate collinear spin density wave (SDW) phase, showing consistency with neutron scattering data for the Fe1.125Te system. Binary FeTe should exhibit superconductivity according to ab initio calculations, but it does not. Different from the computational model, real samples are always non‐stoichiometric (Fe1+xTe). The combined ab initio calculations and Mössbauer spectroscopy of Fe1.125Te reveal two kinds of iron centers forming a plaquette around the interstitial iron ion. This creates a specific magnetic background on which the superconductivity issue should be re‐considered.