The Patent and Trademark Office relied on the provision in the federal statute against registering trademarks that disparage "institutions, beliefs or national symbols."
The Slants won in the Court of Appeals, but agreed that the Supreme Court should take the case because of the importance of the question* (which is the same question that arises in the case involving the Washington Redskins).
The Slants also argue that he government is inconsistent, have rejected and granted trademarks on the names “Heeb,” “Dago,” “Injun” and “Squaw.” They also say that their intent is not to disparage, but to reclaim a term that others have used to disparage — the way some gay people use "queer."
The Redskins make the same argument: They mean the term in a positive way. Whose perspective matters?
* It's also great publicity for the group. They don't need to register their trademark to be permitted to use the name. That's the government's argument: We follow our standards, and we're not going to support what you are doing, but we won't interfere with you. Even if The Slants lose in the end, after winning in the court below, they will have greatly elevated their profile in this world and it will infuse their trademark with more power than any mere registration could impart.