Fe$M_2X_4$ spinels, where $M$ is a transition metal and $X$ is oxygen or sulfur, are candidate materials for spin filters, one of the key devices in spintronics. We present here a computational study of the inversion thermodynamics and the electronic
structure of these (thio)spinels for $M=$ Cr, Mn, Co, Ni, using calculations based on the density functional theory with on-site Hubbard corrections (DFT+$U$). The analysis of the configurational free energies shows that different behaviour is expected for the equilibrium cation distributions in these structures: FeCr$_2X_4$ and FeMn$_2$S$_4$ are fully normal, FeNi$_2X_4$ and FeCo$_2$S$_4$ are intermediate, and FeCo$_2$O$_4$ and FeMn$_2$O$_4$ are fully inverted. We have analyzed the role played by the size of the ions and by the crystal field stabilization effects in determining the equilibrium inversion degree. We also discuss how the electronic and magnetic structure of these spinels is modified by the degree of inversion, assuming that this could be varied from the equilibrium value. We have obtained electronic densities of states for the completely normal and completely inverse cation distribution of each compound. FeCr$_2X_4$, FeMn$_2X_4$, FeCo$_2$O$_4$ and FeNi$_2$O$_4$ are half-metals in the ferrimagnetic state when Fe is in tetrahedral positions. When $M$ is filling the tetrahedral positions, the Cr-containing compounds and FeMn$_2$O$_4$ are half-metallic systems, while the Co and Ni spinels are insulators. The Co and Ni sulfide counterparts are metallic for any inversion degree together with the inverse FeMn$_2$S$_4$. Our calculations suggest that the spin filtering properties of the Fe$M_2X_4$ (thio)spinels could be modified via the control of the cation distribution through variations in the synthesis conditions.
