I think it's time for my obligatory "There's no such thing as chaotic evil or lawful evil, there's just plain evil. Law and Chaos are not and have never been coherently defined in D&D and, realistically, cannot be coherently defined in a system that includes good and evil" post. There are several reasons for this.
One is that "Good" and "Evil" have a special meaning in the English language that is not properly accounted for by the alignment system. In general, good is the opposite of both evil and of bad. Consequently, a concept like the Platonic form of Good or the G.E. Moore's "good" is specific. There is no lawful form of Good and chaotic form of good. There is only the one form of the Good. Similarly, in Moore's system, the optimific action or belief, etc is aimed at a single and simple good. It doesn't come in lawful and chaotic varieties. Under no real-world interpretation of the good would people say "we're both shooting for good but you also want law and I also want chaos." Instead, they would, quite rightly, say that they disagree about either what IS good or how good ought to be achieved or maximized. Quite simply, good subsumes secondary notions and cannot be included in a two-axis system without wrecking it.
Law and Chaos have their own problems, however, independent of the problems created by including good and evil in a two-axis system. Quite simply, the D&D notions of law and chaos lump together a number of various ideas that are actually completely independent and are sometimes even opposed.
For instance, brainstorming for law will usually yield some combination of the following results: respect for the law of the land, respect for authority, organization, civilization, tradition, honor, order, using a system, honesty, reliability, knights, dwarves
Brainstorming for chaos will usually yield some combo of the following results: individualism, free-spiritedness, shamelessness, disrespect for authority, granting no inherent worth to (or sometimes opposing) honor, tradition, and the law of the land, not using a system, flightiness, entropy, barbarians, elves.
All of these are written into D&D in one way or another and, individually, seem to make sense as oppositions. However, combining them into the mega-concepts of law and chaos does not work because many of the individual elements have no connection to each other or actually oppose each other.
Tradition and positive law, for instance, both tend to fall under the auspices of law while individualism and a lack of respect for the rule of law fall under the auspices of chaos. However, tradition and positive law are actually opposing forces. Societies that depend upon positive law to uphold their institutions tend to be highly individualistic (and highly dependent upon the rule of law). On the other hand, societies that lack a formal legal code typically depend upon tradition to uphold their institutions and tend to have a more collective concept of identity. They also tend to be more primitive. Furthermore, honor is a far more important concept in traditional than in modern societies. (This is recognized in fantasy too--how many barbarians have scoffed at the lack of honor among city-dwellers).
So, based on that simple comparison: Tradition (law) goes with collectivism (also law), barbarians (chaos), lack of positive law (chaos), and honor (law). On the other hand, positive law (law) goes with civilization (law), individualism (chaos), non-honor based societies (chaos), innovation vs. tradition (chaos), etc.
If we wanted to base our comparison on the knight/barbarian dichotomy, we would find a different set of paradoxes. The knight supports the rule of law and civilization, and honor but this will often go against established traditions (how many stories tell of knights destroying indigenous cultures--including their own--based upon the edicts of their sovereigns). The barbarian, on the other hand, believes in the traditional ways of his people, personal honor ("you have insulted my honor..."), and living in harmony with the land. In fact, if one goes back to the law/chaos brainstorm, the only real difference between the paragon of law (the knight) and the paragon of chaos (the barbarian) is which lawful concept--the law of the land or tradition--they tend to uphold and whether they represent civilization or barbarism.
At every stage that you attempt to make logical sense of the law/chaos axis, you will find more problems. The easiest solution to this is to simply recognize that the law/chaos axis is a bunch of nonsense--a random group of ideas strung together without any real concern for whether they actually belong together or not.
As commonly used, chaotic evil types tend to be more destructive of society and lawful evil types tend to be more subversive of society. Increasing chaotic evilness risks anarchy while increasing lawful evilness risks tyranny. (These are not hard and fast distinctions just how they're often used in games. There are plenty or arguments for chaotic evil tyrannies and lawful evil creating anarchy--that's because nobody actually uses law and chaos for exactly the same things and the D&D system uses them for a number of things that actually have no real connection). Whether it's better to live in Rwanda or the Stalinist USSR is an open question. My observation is that it seems to take longer to pick up the pieces after anarchy but that tyranny is more likely to endure for several generations. (Then again, Rwanda's been a mess for pretty much as long as North Korea has been a tyranny so maybe that's not right either). They're both bad.
