Note: I'm going to ignore the fact that these parties hardly encompass (look: here and here) the ideas I want to express.
The idea is that the Right reveres freedom above all else (as they point out our founder fathers did) and the Left highlights equality as most important (as became especially clear during the Civil Rights Movement). Case in point: following the Civil Rights Movement, restaurant owners in the South (and everywhere else) did not have the freedom to refuse service to people for arbitrary reasons (like the pigmentation of their skin).
This example may be favorable to the Left's position, but it's the elephant in the room of American history, so I'd deem it fair.
This debate has raged now at least since the 1950s (much longer actually) over things like income taxes, welfare, capital gains taxes, education, unions, environmental policy, progressive vs. regressive tax structures, and mandatory health insurance. So, pretty much everything economic.
I'd like to side with neither completely. In fact, I'd like to say that we need both. We need both as much freedom as we can get our hands on and a completely level playing field (everyone starts out completely equal and lives their lives from there).
Let me make several observations. We could opt for equality without freedom (benevolent dictator anyone?), but we could not opt for freedom without equality. Here's why:
The legal right to do something does not equate to the freedom to do that thing. Freedom is bigger than legal permissibility. It is the aligning of not just legal, but also financial, emotional, psychological, and mental abilities (etc., etc., and etc.). Take the "ability" of African Americans to vote after the 15th Amendment. The legal ability didn't mean much after southern bigots took over the electoral process after Reconstruction. It took until 1965 and the Voting Rights Act for much change to happen and even then it was gradual. These, of course, were both laws dealing with legal ability, but don't miss my point: there were other factors (besides legal) at play that determined the "freedom" to vote.
Many (or most) of the non-legal barriers to freedom (I hope I don't need to enumerate exactly what these are) would be solved by increased equality. So really, we can't have "freedom" in the true sense of the word if we don't have equality.
We must acknowledge that for our society to achieve personal freedom, we must achieve personal equality. Espousing a policy of "freedom" does not give us freedom from worrying about equality. Rather, Freedom is derived from equality.