NEW ROCHELLE, N.Y. — The way Moises Valencia describes things, it began with a simple idea.

Seeing that the American flag flying outside the old military armory in the city needed replacing, he took it upon himself to contact local veterans about putting up a new one. For good measure, he shelled out about $16 online for a yellow Gadsden flag, bearing an image that dates back to 1775, of a coiled rattlesnake and the words “Don’t Tread on Me,” in case the veterans wanted to use that one, too.

The way city officials see things, a group whose agendas go beyond purely patriotic ones decided to use a public space, the flagpole at the city-owned armory, to fly a banner that has come to symbolize the Tea Party and antipathy to government....