  Supernova remnants (SNRs) are a complex class of sources, and their
heterogeneous nature has hindered the characterization of their general
observational properties. To overcome this challenge, we use statistical tools
to analyze the Chandra X-ray images of Galactic and Large Magellanic Cloud
SNRs. We apply two techniques, a power-ratio method (a multipole expansion) and
wavelet-transform analysis, to measure the global and local morphological
properties of the X-ray line and thermal emission in twenty-four SNRs. We find
that Type Ia SNRs have statistically more spherical and mirror symmetric
thermal X-ray emission than core-collapse (CC) SNRs. The ability to type SNRs
based on thermal emission morphology alone enables, for the first time, the
typing of SNRs with weak X-ray lines or with low-resolution spectra. We
identify one source, SNR G344.7-0.1, as originating from a CC explosion that
was previously unknown, and we confirm the tentative Type Ia classifications of
G337.2-0.7 and G272.2-3.2. Although the global morphology is indicative of the
explosion type, the relative morphology of the X-ray line emission within SNRs
is not: all sources in our sample have well-mixed ejecta, irrespective of
stellar origin. In particular, we find that 90% of the bright metal-line
emitting substructures are spatially coincident and have similar scales, even
if the metals arise from different burning processes. Moreover, the overall
X-ray line morphologies within each SNR are the same, with <6% differences.
These findings reinforce that hydrodynamical instabilities can efficiently mix
ejecta in Type Ia and CC SNRs. The only exception is W49B, which is likely from
its jet-driven/bipolar SN explosion. Finally, we describe observational
constraints that can be used to test hydrodynamical models of SNR evolution;
notably, the filling factor of X-ray emission decreases with SNR age.
