Neither the United States nor any of its states prohibit wearing a swastika or carrying it on a banner. Legislatures cannot legally create such a prohibition, consistent with long-dominant understandings of the significance of the First Amendment of the U.S. Constitution.
The First Amendment of the U.S. Constitution reads, “Congress shall make no law … abridging the freedom of speech.” The word “speech” has long been understood to include more than verbalization. This amendment also protects the right to wear or display controversial symbols as part of the general principle that, as the Supreme Court said in 1957, “Every citizen shall have the right to engage in political expression and association." Further, although the First Amendment on its face bars only acts of Congress, it has been applied to the acts of state legislatures because the 14th Amendment, enacted in 1866, says that “no state shall” deprive any person of life, liberty or property without due process of law. The freedom to wear a swastika is part of the liberty no state may infringe upon.
The U.S. Supreme Court confirmed the fact that wearing this symbol is part of the constitutionally-protected liberties of Americans in the case of National Socialist Party of America v. Village of Skokie (1977). Scheduled to take place on Hitler's birthday, Nazis planned a march in which they would wear swastika arm bands and carry flags representing the Nazi era in Germany, to be held in a community that was home to 5,000 Holocaust survivors. After a Supreme Court ruling made it clear that they would be permitted to do so, the marchers compromised and agreed to hold their rally in Chicago's Marquette Park instead.
Parks and Streets
Parks, like streets, consist of public land, and freedom of speech rights have a strong claim there. Also, perhaps more obviously, you may dress up as you please on land you own. But privately-owned land that belongs to somebody else, even if that private land may seem intuitively like a public space, such as the common areas in a shopping mall, is constitutionally a very different matter. The Supreme Court has taken the position that the First Amendment does not prohibit a mall owner from barring the distribution of political leaflets in that mall. The reasoning in such decisions would seem likewise to protect actions by that owner and its agents in excluding swastikas.
Public, Private and Quasi-Public
However, the Supreme Court has also held, since 1980, that states have the authority to adopt through their own constitutions civil liberties more expansive than those conferred by the U.S. Constitution. This creates scope for the policy of the state of Florida, for example. As discussed in the case of Wood v. State, Florida law prohibits a private owner of a “quasi-public” place from using trespass laws to exclude peaceful political activity. Thus, in Florida you might also have a legal right to wear a swastika in a shopping mall, despite the facility's policy.