  We introduce a method for jointly registering ensembles of partitioned
datasets in a way which is both geometrically coherent and partition-aware.
Once such a registration has been defined, one can group partition blocks
across datasets in order to extract summary statistics, generalizing the
commonly used order statistics for scalar-valued data. By modeling a
partitioned dataset as an unordered $k$-tuple of points in a Wasserstein space,
we are able to draw from techniques in optimal transport. More generally, our
method is developed using the formalism of local Fr\'{e}chet means in symmetric
products of metric spaces. We establish basic theory in this general setting,
including Alexandrov curvature bounds and a verifiable characterization of
local means. Our method is demonstrated on ensembles of political redistricting
plans to extract and visualize basic properties of the space of plans for a
particular state, using North Carolina as our main example.
