  We consider the measures of partisan symmetry proposed for practical use in
the political science literature, as clarified and developed in Katz, King, and
Rosenblatt (2020). Elementary mathematical manipulation shows the symmetry
metrics to have surprising properties that call their meaningfulness into
question. To accompany the general analysis, we study measures of partisan
symmetry with respect to recent voting patterns in Utah, Texas, and North
Carolina, flagging problems in each case. Taken together, these observations
should raise major concerns about the available techniques for quantitative
scores of partisan symmetry -- including the mean-median score, the partisan
bias score, and the more general "partisan symmetry standard" -- as the
decennial redistricting begins.
