High pressure can provoke spin transitions in transition metal-bearing compounds. These transitions are of high interest not only for fundamental physics and chemistry, but also may have important implications for geochemistry and geophysics of the Earth and planetary interiors. Here we have carried out a comparative study of the pressure-induced spin transition in compounds with trivalent iron, octahedrally coordinated by oxygen. High-pressure single-crystal M{o}ssbauer spectroscopy data for FeBO$_3$, Fe$_2$O$_3$ and Fe$_3$(Fe$_{1.766(2)}$Si$_{0.234(2)}$)(SiO$_4$)$_3$ are presented together with detailed analysis of hyperfine parameter behavior. We argue that $zeta$-Fe$_2$O$_3$ is an intermediate phase in the reconstructive phase transition between $iota$-Fe$_2$O$_3$ and $theta$-Fe$_2$O$_3$ and question the proposed perovskite-type structure for $zeta$-Fe$_2$O$_3$.The structural data show that the spin transition is closely related to the volume of the iron octahedron. The transition starts when volumes reach 8.9-9.3 AA$^3$, which corresponds to pressures of 45-60 GPa, depending on the compound. Based on phenomenological arguments we conclude that the spin transition can proceed only as a first-order phase transition in magnetically-ordered compounds. An empirical rule for prediction of cooperative behavior at the spin transition is proposed. The instability of iron octahedra, together with strong interactions between them in the vicinity of the critical volume, may trigger a phase transition in the metastable phase. We find that the isomer shift of high spin iron ions depends linearly on the octahedron volume with approximately the same coefficient, independent of the particular compounds and/or oxidation state. For eight-fold coordinated Fe$^{2+}$ we observe a significantly weaker nonlinear volume dependence.
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