>>It depends on the state. In some states you can as in North Carolina. In others you can only vote in the primary for your party candidates. You cannot switch parties or vote outside of it. In others, you can register for any party at the caucus/primary and vote in that party even if the day before you were registered to another party.
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>And if you did, how would they know that you aren't voting in both places? Is there a centralized table with your PK in there? How can that possibly work?

Here in Texas, they used to stamp your voter registration card announcing which primary you voted in.