I've always viewed the Lawful-Chaotic spectra as "following the rules vs. following your heart".

A lawful good character follows the rules because they believe rules serve a greater good

A chaotic good character follows their heart because they believe rules aren't always just. They'll obey the rules only when they agree with them.

Lawful evil follows the rules because it's the least troublesome way of getting/doing what they want, in some cases even the most satisfying - because it's harder to fight back when the law is on your side.

Chaotic evil efficiently follows their impulse. They take and do whatever they want to do, whether they're allowed to or not, because they don't believe in the rules. If the rules allow them to do what they want, that's just a convenient coincidence.

Neutrals are a huge spectra in between. They could a character with their own moral code, I.e. an evil character who refuses to kill kids, or a good character who will always bends the rules to screw over ruch people.

I like to try to give chaotic its own set of things they can all agree on that would make lawful uncomfortable, but that they’ll fight each other over implementation, much like how a lawful good character will agree with a lawful evil character that strong leadership is important and that certain people are more deserving than others and that they need to establish firm rules but then fight over who it should be and what those rules should be.

I think of a chaotic character as someone who may not know what they’ll do until they’re put in a position. They hate hierarchy and being subjected to authority and don’t want to hold positions of authority. They commit all the crimes they’d like to that they feel comfortable committing (for chaotic good that may be freeing slaves or stealing food for the hungry, for chaotic evil it’s more likely to lean evil). They value freedom and pleasure. While lawful and neutral characters are kept evil or neutral by codes and rules, chaotic characters are kept there by the nature of their desires.

Chaotic evil: insane/short sighted evil, this is folks like serial/spree killers, traveling con artists, the joker. If they have an ideal it is chaos or pleasure and any thinking things through is purely to get it. This is the alignment most likely to acknowledge the full breadth of their evil without making excuses. They don’t need a reason to hurt people beyond what they want and they don’t need to excuse it to themselves after.

Chaotic neutral: follow their whims which neither tend towards particularly good nor evil. They aren’t charging into a burning building or laying down their lives for freedom, but they’re also not doing the sort of shit that the average person would feel bad about like rape or murder. They think of chaotic good as foolish and chaotic evil as fucked up. These are the punks who never contribute to mutual aid or seek out Nazis to bash. These are the people who deal weed and psychedelics, but don’t touch meth or opiates. The people who make moonshine. The sort of person who has to learn why a law is harmful before they don’t go out of their way to break it. They may think of themselves as a free spirit or an asshole or something similar.

Neutral evil: self interest above all else. Maybe they have ideals, but they’ll break them for non-extreme reasons but they excuse it. They’re more likely than anyone else to see vulnerability as an invitation. When the navy is strong, they’re in the colonial force of it, when it’s weak they’re a pirate, and when it switches so do they. This is gang member, the embezzler, the oil executive. They may hate tyranny but think there need to be some rules, but they think there have to be exceptions and don’t they deserve what they’re doing. They may think everyone is like them. They’re drawn to evil acts by evil desires and/or evil biases and preconceptions. Bigotry is another trait that may bring an otherwise true neutral person to neutral evil. Lawful evil will likely try to use these people to fill out their ranks.

This makes me think that Lawful Evil and Neutral Evil are way worse people than Chaotic Evil. The Lawful and Neutral ones are systematically and intentionally evil. While the Chaotic ones are just living their best life—which happens to be evil.

The difficult question is if a chaotic evil character would follow a law that is in their self interest.

I don't think they would. I think they'd break the law anyway, even if it went against their own interests, because their nature is pure lawlessness.

Chaos, to me, implies an inherent lawlessness that is itself the character's goal. They break the law for its own sake, because they want laws to be broken (i.e. for the lulz)

True neutral wouldn't harm others for their own benefit. Saying someone would do anything with no regard for morality implies they are willing to harm others.

Good goes out of their way to help others.

Neutral doesn't go out of their way to help, but also doesn't harm others.

Evil is willing to harm others.