  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.
