Yes, we should take a look. This is going to be long, but only because I'm serious about conveying information and so forth. Many of the replies so far are the kind of thing I hear in any other politics discussion, where the people don't have a handle on it because the knowledge is just too vague. That wasn't to slam anyone, it's just an observation. People discuss because they want to find out the root of things, right? You want to know some facts, you ask questions.
My source is a primer, and early on it explains what it calls "essentially contested concepts." I would word that as "the minimal set of assumptions you need to make definitions for a discussion."
First, we need to know what the problems are that these ideologies try to address. So one of the minimal assumptions we need is to pin down the things we want. This is an "essentially contested" bit--it's open to debate, but in the end you may all agree on Freedom, Equality, and Democracy as building blocks for our present society. There are, you may find, 2 options for achieving each of these goals.
Liberty: Positive and Negative. This isn't a value judgement, it's just a label so we all know how we're going to operate.
Positive liberty is a socialistic view, and belongs to the left.
Equality: Process-oriented and Results-oriented. Process-oriented equality means a level playing field. The quickie term for it is strict meritocracy. We all have the same right to work and succeed. Individual differences, advantages, or limitations are ignored. You make it on your own, sink or swim, etc. This is a small-government concept that takes pure negative freedom, and it is right-wing. Results-oriented equality. This recognizes the individual, and strives for equality in the outcome regardless of conflicts in the process. So we get social programs to help people succeed--welfare, scholarships, affirmative action, the ADA, and so on.
Democracy: Elitist and Participatory. Elitist democracy is closest to a republic, which is why I'm guessing they call themselves Republicans. Modern countries are too big to let everyone have a say in every process, and we don't want to be in a town hall to decide whether to let dogs run loose on the beach every day of our lives anyway. So we elect officials who assume the concentration of the power of their constituents. Participatory democracy is the complete opposite, where everyone pariticpates. If you don't vote, that's your vote, and your wiggle room is much less than if there were a margin of thousands. This works on small scales such as neighborhood councils, town halls, etc. and is a good way to get stuff done. For instance, we had a meth house next door and now the place is infested with rats which create a problem for all of us. We don't call the governor to take care of it, so in this instance we have a degree of self-sufficiency provided by basically forming a union.
Now we know the difference at its core. Pure leftist politics takes on Positive Liberty, Participatory Democracy, and Results-oriented Equality, and pure rightist politics demands the other option. We can't have it at the extremes of the spectrum, but together the two get us what we need.