We have studied several synthetic hematite-like materials, produced via different reactions using various hydrothermal conditions and various temperatures of annealing in air, by bulk elemental analysis, weight loss measurements, scanning electron microscopy, powder X-ray diffraction, Mössbauer spectroscopy, and SQUID magnetometry. We conclude that hematite-like materials cannot be related to pure stoichiometric hematite via a single stoichiometric or physical parameter and that at least two degrees of freedom are required. This is most clearly seen when we introduce a plot of the cell parameter c versus the cell parameter a on which hematite-like materials do not fall on a single line but occupy an entire region that is bounded by hydrohematite-hematite and protohematite-hematite lines. A Morin transition boundary on this c-a plot separates a region where Morin transitions occur from a larger region where Morin transitions do not occur down to 4.2 K. Previous claims that particle size is the dominant factor controlling the Morin transition are understood in terms of correlations between stoichiometry and particle size that are produced at synthesis. Changing contents of incorporated molecular water and structural hydroxyls with associated cation vacancies have different characteristic effects on the crystal structure and move the sample coordinates in different directions on a c-a plot. It is also shown that an accessory sulphate content is adsorbed on the individual hematite crystallites and is not structurally incorporated. Mössbauer spectroscopy is used, as usual, to identify and characterize the spin structure. In addition, hyperfine field distributions from room temperature spectra, extracted by a new method, give a sensitive measure of sample conditions but not a unique one since several factors affect the extracted distributions in similar ways.