But this isn't how Law and Chaos work in D&D.
But it isn't how it works out in the real world either. The CG and the LG may have some measure of mutual respect for each while each feeling the other is unwise and corrupted by a flawed ideology, and they may stay out of conflict with each other - until you put those values in conflict. In the same way, the soldier and the pacifist may respect each other right up until you put those values in conflict. I mean, haven't you ever seen the movie 'Sergeant York'?
At some point, people with strongly differing values do come into conflict in the real world. And at some point in the real world, when you put them under pressure, individuals and societies do invoke the 'enemy of my enemy' principle and decide which values they are going to compromise on. "Sure, he may be a thug, but at least he isn't communist..." or "Sure, he may be a thug, but he's fighting Hitler...", or whatever. Maybe law and chaos aren't the best axis to describe the tension of these choices, or maybe the real world has multiple axis or none, but that doesn't mean that in are fantasy setting the outcomes of assuming the law/chaos axis is part of the social understanding (and indeed is a real and tangible thing) produces results in the social or political spheres that are unbelievable.
The LG person doesn't take the view that the CG person is simply choosing a different value to pursue out of some range of equally good (or perhaps incommensurably good) options. The LG person regards the CG person as morally flawed.
To make this really personal, in the case of something like pacifism, I do consider this a sort of moral failing. Evil has to be confronted. You confront evil on the basis that the person is your fellow human, and with compassion for that person, which strongly moves one toward preferring a pacifist approach. But by taking an absolute stance against violence, you are at some point enabling evil. Further, you have ceased to choose life over death and instead have chosen the death of the nonviolent as preferable to the death of the violent. Pacifist societies exist at the sufferage of the violent societies that enclose them, tacitly allowing others to use force on their behalf so that they can maintain a false righteousness. I have great respect for the intentions of my pacifist brothers, but ultimately if you look at a figure like Ghandi and where advocating utter pacifism takes one morally, I consider it not only morally flawed but a very dark place.
So in the same way I can see this in myself (and also see that others my strongly disagree for reasons I find sympathetic and comprehensible), I don't find it hard at all to imagine the dynamics between LG and CG, or LG and LE. For example, I think you can see in analogy dynamics between LG and LE or CG and LG when you consider the respect figures like Robert E. Lee or even Erwin Rommel invoke even in (and maybe especially in) people who abhor the values that they were ultimately defending. This is particularly interesting in the case of say Robert E. Lee, whose 'lawfulness' (as it were) seems to have motivated him to defend an intuition (slavery) that he himself found morally repulsive.
Heck, in Planescape at least I think it's expected that the LG person might ally with the LE person to fight the CG person.
That makes no sense. It can't be both that the LG person regards the CG form of life as fully permissible, and regards the CG form of life as needing to be opposed.
I didn't say that the LG regards the CG form of life as fully permissible. I said that LG and CG could both find common values AND sharply disagree. So my question to you was, "If we can both have common values and sharply disagree IRL, what is unbelievable about representing morality with two axis?"