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Mr. Haynes said that accepting the name of Freedom Party was a “prudent move” and a “wise thing to do to try to mitigate some of the potential for controversy and offense.”

“This is a story about how the First Amendment sometimes protects speech that people find offensive,” he said. “The Department of Transportation has done its best [and] worked to protect the rights of people, at the same time as trying not to offend citizens as much as possible.”

Mr. McBride and Ms. McManus’ request has been viewed by some as a publicity stunt. A DelDOT official tried to steer Mr. McBride away from participating in the program in an Aug. 25 letter that was released Friday.

“Of course, you remain free to exercise other opportunities to seek publicity for this entity and your litter-control activities, in ways that do not suggest that the state is endorsing this entity,” Joseph Wright, DelDOT maintenance and operations director, wrote. “For example, you have ready access to social media on the Internet and other ways of publicizing your activity, without involving the use of the official Adopt-A-Highway signs.”

Neither Mr. McBride, a 24-year-old resident of Lewes, Del., nor Ms. McManus returned phone calls or emails seeking comment. According to the local press accounts, the group believes in the supremacy of white Christians and that other races and religions should be forced to leave the country.

Jason Hiecke, the chief of state for the National Socialist Movement, emphasized that Mr. McBride, his wife and other members of the National Socialist Freedom Movement Nazi Party are former members of the larger national group and that his organization is not affiliated with the Delaware group.

Mr. Hiecke said that while he can understand the state’s concern of being associated with the organization and agrees that the attempt to have Nazi Party written on the sign may have been a cry for attention, “anyone that wants to help clean up their community should have a right to.”