Just some semi-random thoughts provoked by Aberzanzorax:
The LG paladin holds to a creed that denies many opportunities for personal reward. In the world, holy powers are granted to ensure the success of the paladin's mission. In the game, holy powers can be seen as a compensation for the temporal sacrifices the paladin makes to follow and exemplify the LG creed.
(Aside: We may argue that NG is "purer" good than LG, but an alignment with a N component is by definition one that allows for some flexibility in the face of difficult choices. Flexibility is not a bad thing, but "serve the greatest good in whatever manner is expedient" is easier than "serve the greatest good while upholding lawful and just government". In situations where a lawful government is making an unjust choice, the NG champion can simply rebel, whereas the LG paladin would find a way to root out the evil within the government, if possible, to turn it back to the good.)
So if we were to have paladins by creed, I would hope that alternative creeds could lead to in-game situations requiring the paladin to take a very difficult road to uphold the creed, at the cost of their personal wealth or honor.
Are the other corner alignments capable of supplying such a creed?
CG: The greatest good is personal freedom. These are the charitable libertarians. They want to do good, but respect your right to abstain. And since their own personal freedom is an example of the greatest good, what is there to prevent them from acting as they wish, as long as they do good? I don't see a difficult creed here. Anti-slavery and anarchism might be components of a creed with a cost, but only in a game world where LE governments and slavery are common.
CE: This is the alignment for sociopaths and solipsists. Only their own desires matter. Even if they are pledged to a demon lord, they will do whatever they can to enrich themselves within the scope of their bonds, and may struggle against their bonds and betray their masters. Again, not a creed requiring personal sacrifice.
LE: Respect the hierarchy, whatever the cost to yourself. If you don't like being on the bottom, then claw your way to the top. A LE character craves power and will do anything to get it. What does it mean for a PC to follow the creed of LE? It means obeying the orders of your superior, or your supernatural patron, without question. And since you're evil, you have no remorse--you'll do what you're told, whether it's planting a garden or razing an orphanage.
Your interpretation of the corner alignments may not be the same as mine. But I don't see any of them as supplying a creed comparable in difficulty to the paladin's, which puts the well-being of others always above their own, and so often forces them to take the more difficult road to their goal. Yes, it's a pain in the butt for the paladin's companions; which is why it's a class that should be taken only if the rest of the group is on board with playing heroes.
I'm also thinking about what it might mean to have a paladin of pure law or pure chaos, in the Moorcock sense. Binding yourself to a supernatural patron to gain powers in exchange for pursuing their agenda sounds like a cleric to me, not a paladin. (Marching orders from Arioch!) What is the creed that would call a human warrior to serve Law?
Frankly, I have no idea. I'm very interested in yours!