The centroid energy of the Fe K$alpha$ line has been used to identify the progenitors of supernova remnants (SNRs). These investigations generally considered the energy of the centroid derived from the spectrum of the entire remnant. Here we use {it XMM-Newton} data to investigate the Fe K$alpha$ centroid in 6 SNRs: 3C~397, N132D, W49B, DEM L71, 1E 0102.2-7219, and Kes 73. In Kes 73 and 1E 0102.2-7219, we fail to detect any Fe K$alpha$ emission. We report a tentative first detection of Fe K$alpha$ emission in SNR DEM L71, with a centroid energy consistent with its Type Ia designation. In the remaining remnants, the spatial and spectral sensitivity is sufficient to investigate spatial variations of the Fe K$alpha$ centroid. We find in N132D and W49B that the centroids in different regions are consistent with that derived from the overall spectrum, although not necessarily with the remnant type identified via other means. However, in SNR 3C~397, we find statistically significant variation in the centroid of up to 100 eV, aligning with the variation in the density structure around the remnant. These variations span the intermediate space between centroid energies signifying core-collapse and Type Ia remnants. Shifting the dividing line downwards by 50 eV can place all the centroids in the CC region, but contradicts the remnant type obtained via other means. Our results show that caution must be used when employing the Fe K$alpha$ centroid of the entire remnant as the sole diagnostic for typing a remnant.
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