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Perovskite oxide heterostructures host a large number of interesting phenomena such as ferroelectricity, which are often driven by octahedral distortions in the crystal that may induce polarization. SrHfO3 (SHO) is a perovskite oxide with a pseudocubic lattice parameter of 4.08 Å that previous density functional theory (DFT) calculations suggest can be stabilized in a ferroelectric P4mm phase when stabilized with sufficient compressive strain. Additionally, it is insulating and possesses a large band gap and a high dielectric constant, making it an ideal candidate for oxide electronic devices. To test the viability of epitaxial strain as a driver of ferroic phase transitions, SHO films were grown by hybrid molecular beam epitaxy (hMBE) with a tetrakis(ethylmethylamino)hafnium(IV) source on GdScO3 and TbScO3 substrates. Strained SHO phases were characterized using X-ray diffraction, X-ray absorption spectroscopy, and scanning transmission electron microscopy to determine the space group of the strained films, with the results compared to those of DFT-optimized models of phase stability versus strain. Contrary to past reports, we find that compressively strained SrHfO3 undergoes octahedral tilt distortions without associated ferroelectric polarization and most likely takes on the I4/mcm phase with the a0a0c– tilt pattern.