Gay and Republican can intersect. A gay Republican organization is called the 'Log Cabin Republicans," a name with typical gay innuendo attaching.... here's their site:
http://www.logcabin.org/
There has got to be more dissonance than ever these days between a gay and a Republican identity. But I have family history on how 'gay' and 'traditional respectable' definitively could go together, and lead a gay person to disapprove of a 'long-hair' relative back in the 70's.... Identity is not a thing that always makes logical sense in all its parts.