If the party can punish states for moving their primary “too early”, why can’t it punish them for allowing cross-over voting?
Alternatively, why not just declare an end to the Stalinist practice of giving special consideration to Iowa and New Hampshire, and proceed towards and actual democratic primary system?
Or better yet, let FreeRepublic members pick the Republican nominee. No kidding.
The rules and regulations for the primaries are not entirely up to the state parties. The governments of the states have some part in them as well.
In Texas you can vote in either primary, but only one of them. They must be held on the same day, and as far as I've experienced in 34 years in Texas, they have their polling places collocated, be they in a school or a church.
I'm sometimes surprised that the roof doesn't fall in on the DemonRat polling place. Other times I'm surprised that the Powers that Be allow voting in churches, be it for primaries, general elections or local bond elections. Of course I cast my first Presidential votes in a church, Methodist IIRC (and I do, it's across the street from a "Wesleyan" University). Voted for Richard Nixon. :)