Touchy subject, I agree. Being the Grand Dragon of the local KKK chapter might be legal, but I would not want that person in my place of employment. A loyal Antifa member? Same deal.
I also agree with Old Enough To Know and B....Most companies have policies/codes of conduct that address such activities in order to maintain a non-hostile work environment. In addition, employment is "at will" in most states.
How far could an employer take this? What is unsavory? Eating meat is unsavory to a vegatarian or a Hindu, would posting recipes of pulled pork be offensive/unsavory? A very slippery slope, as commented on earlier!
My gut reaction to the question, as written, is yes, you should be able to be fired for unsavory activity. Exhibit A: Roseanne Barr.
However I voted no in the poll because your question omitted a key detail: What are the terms of the employment contract? A smart company will put these provisions in writing to account for this possibility. If that language is not present, then I don't believe the company has any legal or ethical grounds to terminate employment.
I agree with "Old Enough to know"
Whose definition of unsavory? This could be a very slippery slope.
It does need to be proved true, but if it is, then yeah, you have an interest in maintaining your working environment even if you don't go for the moral responsibility to your other employees. If someone is spreading literal Nazi propaganda, I'm not going to tell their Jewish coworker that it's just on the weekend, oh well.
"Unsavory" is a bit of an understatement when these philosophies are, by definition, literally about promoting violence against specific people. Feeling unsafe does not a productive company culture make.
Sure-- assuming there is a morals clause in the employment agreement or the company's code-of-conduct addresses this. Lets also not forget that most states employment is "at -will states. And... if treated unfairly, the ex-employee or ex- employer can respond through many other pathways
Another form of lynch mobs for unproven rumors - just look to India to see the chilling effect this had