Why do so many people see gays and Republicans as polar opposites? For starters, we are.
One year, when I gave the World's Smallest Political Quiz at a Gay Pride festival, the people who took the quiz typically scored at least 70% on personal liberty but were evenly distributed on economic liberty. In other words, they formed a band between the liberal and libertarian corners of the Nolan chart.
By contrast, a defining characteristic of a Republican is opposition to personal liberty. Republicans can be anywhere on economic liberty, although I suspect that they are increasingly opposed to that as well. That is, they should form a band between the conservative and authoritarian corners of the Nolan chart.
A graphical representation of what I mean would look approximately like this:
As you can see, gays and Republicans are, for the most part, staring at each other from opposite sides of the Nolan chart.