Almost every President going back to Franklin D. Roosevelt, and many before him, has chosen cabinet-level members from the other party–though in quite a few cases the party identification was fairly nominal. Of course Rs tended to pick conservative Ds, and Ds picked moderate and liberal Rs (as long as there were any).
Here's a list from Wikipedia:
https://en.wikipedia.org/wiki/List_of_United_States_political_appointments_across_party_lines
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@DavidSalo so i guess the question is, in which position could a kinzinger-style republican do the least damage?
she can appoint a republican to be secretary of balloon doggies.