Let's take a look at the political spectrum. The Left-Right spectrum is determined by a number of factors. The major ones are as follows:
Economic Policies (interventionism vs. laissez-faire)
Bureaucracy Policies (large government vs. small government)
Employment Policies (workers' rights vs. employers' rights)
Equality Policies (equality of outcome vs. equality of opportunity)
Religious Policies (secularism vs. religiosity)
Tradition Policies (innovation vs. traditionalism)
Cultural Policies (law dictates culture vs. culture dictates law)
Beliefs of Human Nature (malleable vs. fixed)
Okay, now that we have those up, we need to recognize that the Democratic and Republican Parties are (supposed) to represent the left and right, respectively. So you can't just say, "Well, the Republicans are moving right because they support tax cuts!" because that's what they're supposed to do. If you want to prove they're
moving right, you have to show that (as an example I pulled out of thin air that isn't necessarily true) the Republicans have been cutting taxes even more virulently today than they were twenty years ago. If you want to prove they're moving to the right you have to prove there's more of religious connotation to the GOP now then there was twenty years ago. You get the picture.
EDIT: If anyone's curious, I got the thing on Left-Right politics off Wikipedia. Here's the link.
http://en.wikipedia.org/wiki/Left-Right_politics#Definitions