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I am thinking that one possible asymetry between "the left" and "the right" is that the former is a rather homogenous group, while the latter is heterogenous. The left generally means socialist(-ish), and the right generally means non-socialist. The left is a fuzzy blob in the concept-space, the right seems like a label for points outside of this blob.
As an example, both Ayn Rand and Chesterton would be examples of "the right". What exactly do they have in common? (Religion: the best thing ever, or the worst thing ever? Individual or community? Mystery or reason? The great future or the great past? Selfishness or selflessness? Should women be allowed as leaders? Etc.) The common trait that classifies them both as "the right" is the fact that neither of them is a socialist.
Well, I could also says that neither of them "considers hinduism the best thing ever"... but why should that information be used to classify them? Well, for a hinduist that would be an important information. Then it follows that classifying many diverse views under one label of "the right" makes sense to you mostly if you are a socialist. (Or if being versus not-being a socialist is the dominant question in your political paradigm.) An "Ayn-Rand-type" non-socialist and a "Chesterton-type" non-socialist would otherwise feel uncomfortable under the common umbrella.
I am not saying there are no differences among "the left", but to me they seem more like a matter of degree. This observation may be culture-dependent. I am from eastern Europe, where "the left" basically either wants "what communists did" or "something similar to what communists did, just less, and if possible without all the violence". -- I suppose in USA the diversity of "the left" is greater, because there is no such attractor. Actually, the Republican party may serve as a similar (though weaker) attractor for "the right".
OK, what I tried to say was this: suppose that the leftist opinions are pretty similar, and the rightist opinions are very diverse. Assuming that both sides are about equally mindkilled (believe in about the same proportion of true statements, and the same proportion of false statements), for most statements the left will probably have either true believes or false beliefs as a whole, while the right will internally disagree -- therefore even if each specific right political group has the same chance to have true beliefs, there is a very high chance that at least one of the right political groups will have a true belief.
For clarity, here is a model: There are true beliefs A, B, C, D; and every political group is correct only about one of them, and incorrect about three of them. There are three left groups, but all of them believe in A. There are three right groups, first of them believes in B, second in C, third in D. -- Now if we make a per-group statistics, we find that each party is 25% correct and 75% incorrect. However, if we make a per-true-belief statistics, we find that 25% of true beliefs are associated with the left (A), and 75% of true beliefs are associated with the right (B, C, D). -- In this model, if a group of people could succeed to hold all true beliefs (A, B, C, D), an external observer would judge they are mostly right (despite they happen to disagree with every individual right group in majority of beliefs).
Back to the beginning -- we disagree with Ayn Rand about simplicity of values, or about importance of community; we also disagree with Chesterton about religion. That alone does not give us a political label. On the other hand, disagreeing with a socialist political idea is sufficient to get the label of political right, because any point outside of the socialist concept-space is called "the right".
Beware the out-group homogeneity effect. People tend to see their own group as more heterogeneous than other groups, as differences that look small from far away look bigger up close.
With left and right, I have also heard the exact opposite claim: that the "right" represents a narrower, more coherent group. In the US, the "right" is based in the dominant, mainstream social group (sometimes called "real America"), drawing disproportionately from people who are white, male, Christian, relatively well-off, straight, etc., while the "left" is a coalition of the various groups that are left out of "real America" for one reason or another. Alternatively, conservatives are the people who support the existing social order and want to keep things roughly how they are; liberals are the ones who want change - and there are more degrees of freedom in changing things than in keeping things the same.
Seems to me there could be a common pattern:
If a political group X, identified as A in {"left", "right"}, becomes very powerful in some era, the following things happen:
Later, when the political group X loses some power:
In USA, X = Republican / religious right, and A = "right". In Eastern Europe, X = Communist, and A = "left".
This is very simplified, but it explains why sometimes the same person could self-identify as "left wing" in USA (to express their incompatibility with the religious right), and as "right wing" in Eastern Europe (to express their incompatibility with the communists). On the other hand, people mostly compatible with the religious right or with the communists can self-identify the same in both places.
In Eastern Europe the distinction between "support the traditional model" and "support change" is rather confused, because it is not clear whether the traditional refers to the era before the fall of communism, or to era even before the communists. In some sense, both religious right and communists are literally the conservative parties here.