For bikers, the patch is key to belonging and the optics of appearing tough, and members can spend months or even years proving themselves before they earn the right to wear it.

“The patch is like the American flag to these guys and speaks to the identity of the club, the individual and the culture,” said William Dulaney, a retired associate professor who is an expert on motorcycle groups. “Some clubs have the rule that if the colors even touch the ground, they have to be destroyed.”

The Mongols’ marks, like those of other biker groups, are registered with the United States Patent and Trademark Office. Clubs have aggressively protected their patches from unauthorized use.

The Hells Angels have gone after large corporations including Toys “R” Us, the Alexander McQueen fashion line, Amazon, Saks, and Walt Disney, accusing them of infringement on its death’s head logo — a skull in a winged helmet — and other club symbols.

They have usually been successful, reaching settlements that require defendants to cease using the trademarks and to recall and destroy merchandise, among other concessions.

The Mongols have had their share of run-ins with the law. The group was founded in Montebello, Calif., in 1969 and has about 1,000 members in the United States, most of whom are Hispanic. About half of the club’s membership is in California, though Mr. Santillan said 11 new chapters were recently established in Texas.

In 2012 Christopher Ablett, a suspected member of the Modesto, Calif., chapter, was sentenced to life in prison for the 2008 murder of the president of the San Francisco chapter of the Hells Angels, Mark Guardado. In 2014 David Martinez, a Mongols member in San Gabriel, Calif., was charged with murder in the shooting death of a Pomona police officer.