Police departments across the country have suspended officers or referred them to internal reviews for attending the events on Jan. 6 in Washington that devolved into an assault on the U.S. Capitol.

The commanding officers or officials involved in the cases in New Hampshire, Pennsylvania, Texas and Washington State stressed that while the officers attended as private citizens, the question of whether they broke the law would be investigated.

In San Antonio, Sheriff Javier Salazar of Bexar County said he had referred Lt. Roxanne Mathai to the internal investigations department after she posted a video of herself on Facebook from near the Capitol, wearing a red, white and blue face mask and wrapped in an American flag. The officer waxed enthusiastic about the day but stated explicitly that she would not enter the Capitol.

Plumes of tear gas waft in the background of the video, and Mr. Salazar said investigators would determine whether police had declared the gathering an unlawful assembly. “If that is the case and she remained on scene and began filming and began making challenging statements, that means breaking the law,” the sheriff said.

Mr. Salazar, noting that Ms. Mathai had already been suspended from the force since October on another matter, said at a news conference on Friday that he had referred the video to the F.B.I. and to the internal investigations department.

Ms. Mathai could not be reached for comment.

Thomas Goldie, a Pennsylvania police officer, posted pictures of himself on Facebook from the rally wearing a Trump cap, but there was not yet any indication that he had been in the Capitol, said Jim Miller, chief of the Zelienople Police Department, a 10-member force in a small town 29 miles north of Pittsburgh.

“Him being there is not a problem — he had a right to be there, but not to break into the Capitol, obviously,” said Mr. Miller, adding that the legal department was reviewing the matter. Mr. Miller said Mr. Goldie was on vacation and had yet to return. Mr. Goldie did not respond to telephone messages.

In Seattle, the Police Department announced on Friday that two officers were being placed on administrative leave after attending the demonstration. Internal investigators would determine if the officers were directly involved in any of the illegal events, Chief Adrian Diaz said in a statement.

And in Troy, N.H., Richard Thackston, the head of the town Board of Selectmen expressed support for the police chief, David Ellis, after there were calls for his resignation for attending the events in Washington. Neither man responded to telephone messages left at the town hall and with the police dispatcher.

Mr. Ellis told New Hampshire Public Radio that he opposed the violence. At a regular meeting of the selectmen on Thursday night, Mr. Thackston said Troy residents and many outsiders had called for Mr. Ellis to be dismissed.

He said that while the events at the Capitol were appalling, everybody has the right to participate in political events without fear of being fired. “So, Dave Ellis in my book is just fine,” Mr. Ellis added. “And the rest of the world needs to go about and mind its own damn business.”

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