Hitchens was never a right-winger-he just agreed with neocon foreign policy views. He certainly wasn't a right-winger on social issues and not much of one on economic issues either.
"Not much of one" is an understatement. The guy was a Trotskyist.
I really don't understand how an atheist Trotskyist neocon could win a vote over anyone, at least not someone on the fringe of mainstream.
His argumentative and contrarian demeanor made him a compelling person to listen to, every discussion he participated in was interesting.
Also it hasn't been mentioned in this thread but he gave up on being a socialist after the Cold War (saying that it hadn't done anything to innovate or reinvent itself), but he still considered himself a Marxist and held Trotsky in high regard.
I'm also not sure "neocon" is the best way to describe him because he hated Reagan (he also hated Bush 41 and Clinton but liked Bush 43 and Obama) even after he left socialism.