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The following article from CNN describes a Michigan police officer being put on administrative leave for having KKK material at his home: https://www.cnn.com/2019/08/10/us/michigan-officer-placed-on-leave-kkk-document-house/index.html. The materials were discovered while a potential buyer was touring his house.

Although I vehemently condemn the KKK, doesn't this officer have the right to display whatever he wants in his home so long as it doesn't actively and deliberately call for violence? Aren't these articles protected under the first amendment? I realize this is an extreme example, and as a police officer his job requires interacting with all races, but unless it can be shown that he's bigoted and that it negatively affected his job performance, isn't it illegal to fire him?

Employers can restrict speech according to company policy while at work, but we all have to go home at some point. Can those restrictions follow us after clocking out?

In general, employers are free to fire you for your speech. The First Amendment does not apply to anyone except the government (other than a narrow set of circumstances where private parties act on behalf of the government or take on government roles, like when private universities employ campus police). If the officer was being fired from a job at a private company, this would not be an interesting question -- the answer would clearly be "no, there is no First Amendment claim here."

What makes this interesting is that the government is involved. Unlike private employers, government agencies are bound by the First Amendment. In Pickering v. Board of Education, the Supreme Court held that this does restrict them in their role as employer and that they can't necessarily fire an employee for speech. Pickering imposes a balancing test, where the harm to the employee's First Amendment rights is weighed against the government's interest in efficient operation. Courts have given particular leeway to police departments punishing speech that would undermine public trust and confidence in the department. A police officer expressing racist views, even privately, can seriously hamper the effectiveness of the department if the speech gets linked back to them. For instance, see Pappas v. Giuliani, where the Second Circuit upheld the firing of an NYPD officer for anonymously mailing racist diatribes from home in his off-duty time.

Another answer suggests that the main question is a public safety one: whether the officer could be trusted to carry out his duties without bias. But that's not the only legitimate consideration for the government employer. The courts have repeatedly held that public perception of an agency is a legitimate concern, especially when it comes to agencies (like the police) whose job requires maintaining good relations with the community. In Pappas, the officer was assigned as a computer operator who had no contact with the public, but he was still a police officer whose speech had a high potential to undermine NYPD community relations.

A private employer absolutely can fire you for your speech, even speech off their property, as the First Amendment is a directive to the government. States can pass their own statutes to restrict termination for speech, and some may have.

For the government, the answer is generally "no". But there could be a public safety exception in this situation, where the officer can no longer be trusted to carry out his duties without bias.

The freedom of speech only protects you from the government prosecuting you for expressing your political opinions.

Even if your employer was the government, being fired it not persecution by the government. Getting tried in a court of law is persecution by the government.