It is just chance that the liberals got stuck defending abortion and the conservatives got to attack it. It could have been the other way around. I remember in the 1970s many Republicans supported abortion rights because it fit their view of getting the government off of the backs of individuals (both father and son Bush supported abortion rights in the 1970s). Some liberals thought that the little fetus was a creature we must protect and defend like anyone else. But the NOW changed everything. They insisted on abortion rights because they didn't want society dumping on them. Their view on abortion was the NRA's view on assault rifles - they wanted the option and they weren't going to let politicians take away that option. The NOW was strong in the Democratic party - the rest is history.
Well, they can't switch places on abortion now, but what about the privatization of social security? Arnold Kling explains in Tech Central Station "Why the Left Should Favor Social Security Privatization (and the Right Should Oppose It)".