Whereas I can conceive the various factions of Law working together in a dire situation, Chaotic Good in particular would never ally with Chaotic Evil, which values the diversity that Chaotic Evil destroys senselessly and which Law destroys via a kind of cultural imperialism.
I agree with you re Chaos, but think the same sort of rationale applies to Law.
I'll set out my reasoning: it's not identical to yours but I think there's meaningful overlap.
Because I follow Gygax's presentation of alignment in his PHB and DMG, as the only coherent versions I know of 9-point alignment, I'll be working from those.
And I always start with Good:
good means respecting, advancing and ensuring access to valuable things - life, beauty, truth, freedom, the possibility of happiness. Conversely,
evil doesn't care about any of those things and will happily run roughshod over them to satisfy will and desire (especially the desire for power).
CG thinks that the best path to those values is individual self-determination/self-realisation.
LE agrees - that's why they reject individual self-determination and self-realisation, as they wish to harness everyone for their own self-aggrandisement (ie "to impose their yoke upon the world"). LE takes the view that permitting people to pursue their own good as CG does is an obstacle to the LE goal of power, because it breaks down the necessary hierarchies and structures that LE exploits.
LG, on the other hand, thinks that the best path to achieving and protecting the values of life, truth, beauty etc is through social organisation and respect for tradition. Individualism is a threat because it breaks down the social structures that - LG asserts - make the good achievable.
And CE agrees - that's why CE people assert their individualism and scorn social organisation and tradition, because they see those as obstacles to their own self-aggrandisement.
So I agree that CG and CE can't cooperate, because they value individualism for exactly opposite reasons - CG because they see it as the best way for everyone to live well; CE because they see it as a way of pursuing their own ends without being hindered by things (like structures and traditions) that only matter because they help everyone live well.
But for the same reason I think LG and LE can't cooperate, because they value social structure for exactly opposite reasons - LG because they see it as ensuring that everyone can live well; LE because they see it as a way of building up power and domination without worrying about (what they believe to be) the pernicious effects of social structures on people's wellbeing.
A final comment: as I've just presented it, LG (and CE) and CG (and LE) disagree primarily not on a question of value, but on a question of fact (the facts in question being the nature and consequences of social processes). LGs think that structures produce wellbeing, and that because individualism undermines structures it threatens wellbeing. CGs disagree - they think that individualism produces wellbeing, and hence worry about structures that stifle individualism.
Both can't be right.
And CE and LE mirror this: CE cheerfully break down structures because they agree with LG about the rationale for structures (ie they're tools for spreading wellbeing) and reject that rationale; whereas LE agree with CG about the rationale for structures, but don't care about others' wellbeing and so cheerfully build up the structures that they believe will give them power. Again,
both can't be right.
Anyway, I think this sort of approach to alignment - which brings out an actual, concrete disagreement about something pretty important (ie what is the effect on wellbeing of various social processes and social arrangements that human beings might participate in) - is more interesting than alignment as a personality shorthand.