Reagan was a democrat and later admited he changed his views on abortion. I can't speak for Nixon because I don't remember where he stood on that issue. Rudy admits he supports abortions still, so there is a difference between the two.
You know NY politics better than I do. What you said about party endorsements is similar to what someone said over on the youtube thread. Another poster answered by saying this:
That's true in a sense, but it overlooks the larger picture. After all, Giuiani did not run for, nor did he receive, the nomination of the NYC Democratic Party.
If I'm correct, Giuliani never said he wouldn't run for the Republican nomination, he merely said he'd be willing to accept the Democratic nomination if the situation presented its self.
Clearly, given his views and given his association with Reagan and such, he wasn't going to get anywhere with that, so he didn't run for the Democratic endorsement. But with a system like NY's, you'd have to be either a fool, or an uncompromising ideological purist, not to accept the nomination of any party if it presented it's self as a realistic possibility, because it's virtually impossible for it to hurt you and will all but certainly help you win.
Ed Koch didn't run for the Republican nomination during his first run, but accepted it during his second run. It didn't make him a Republican by any real standard, but it he won with 80something% of the vote because of it.
It also has to be put in the context of Giuliani's support of non-partisan elections in NYC. I'm not sold on his position, but it's a position he's taken for a long time, which further explains why he'd accept the Democratic nomination.
Think of it this way: Would anybody around here complain if Zell Miller were nominated by both the Republican and Democratic parties for President? (no, that won't ever happen, that's not the point) Didn't think so.