Reverse parties on abortion

Starting at Roe v Wade, make the Democrat Party the normally pro-life one and the GOP the normally pro-choice one.
 
Are the Democrats socially liberal on all the other usual issues? If so, it's hard to see why they'd make an exception for abortion.

Maybe if the Republicans had somewhere along the way gone hyperlibertarian, supporting 100% unfettered abortion access as a matter of bodily autonomy, and the Democrats were moderately pro-choice, along the semi-restrictive lines of Roe V. Wade(more or less OTL), the Republicans would end up, by default, as relatively more pro-choice than the Democrats.

But it's gonna be hard to get a libertarian GOP post-1900(and no, Goldwater '64 does NOT make that happen), and in any case, not all libertarians view the bodily autonomy of the mother as trumping that of the fetus(Ron Paul is anti-abortion, for example). But that's the best I can come up with.
 
And if you're willing to have the Dems be SoCon on the issues in general, I guess the question becomes something like "How can we keep the Solid South going?"
 
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