Yes, but (IANAL) if the employee suspects that there was an illegal reason behind it, they can probably sue and subpoena documents revealing management motivations.

Like, sure, a manager could wake up and just fire you because they were grumpy. But if it is discovered that there was internal communications about their political opinions leading up to it, then there would be a case.

You can be fired for your political opinions.

There are certain legal limitations to that from the NLRA. tptacek goes into that in depth here: https://news.ycombinator.com/item?id=16396827

Yes, but there are still prohibited reasons for firing. Which labor organizing is one of; Damore charged that he was fired for organizing, the NLRB found that Google fired him for actions that fall outside of protected labor organizing.

Yes, but not if you're trying to organize to improve working conditions, because this is protected by the NRLA.

If the NRLA believes only parts of Damore's memo were working towards improving working conditions and other parts were working to discriminate against women, which part is which.

To someone that agrees with Damore, it looks like everyone mischaracterized his effort to improve working conditions as sexism. Then Google was able to use that as cover to fire him for protected organizing.

In an at-will state, I can fire you because I don't like how you did your hair, but I can't fire you because you belong to a protected class.