The basis for the legality of wearing an armband to school to protest an action taken by the government, as established in the Supreme Court case Tinker v. Des Moines Independent Community School District (1969), is protected symbolic speech.
In this landmark case, the Court ruled that students do not lose their First Amendment rights to freedom of speech when they enter a school. The armbands in question were deemed a form of symbolic speech, which is protected under the First Amendment as long as it does not cause a substantial disruption to the educational environment. The Court emphasized that schools cannot censor student expression unless it interferes with the operation of the school.