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We report on the chemical structure and spin Hall magnetoresistance (SMR) in epitaxial $\alpha$-Fe$_2$O$_3$(hematite)(0001)/Pt(111) bilayers with hematite thicknesses of 6 nm and 15 nm grown by molecular beam epitaxy on a MgO(111) substrate. Unlike previous studies that involved Pt overlayers on hematite, the present hematite films were grown on a stable Pt buffer layer and displayed structural changes as a function of thickness. These structural differences (the presence of a ferrimagnetic phase in the thinner film) significantly affected the magnetotransport properties of the bilayers. We observed a sign change of the SMR from positive to negative when the thickness of hematite increased from 6 nm to 15 nm. For $\alpha$-Fe$_2$O$_3$(15 nm)/Pt, we demonstrated room-temperature switching of the N\'eel order with rectangular, nondecaying switching characteristics. Such structures open the way to extending magnetotransport studies to more complex systems with double asymmetric metal/hematite/Pt interfaces.