There are generally three types of National Guard deployments.

Troops can be called to active duty by governors in a state duty status, by presidents in a federal status or in a hybrid role known as Title 32, named for a Guard-related section of the United States Code intended for homeland defense.

Mr. Trump’s National Guard mobilization on the border is a Title 32 deployment, in which the soldiers are under the command and control of their governor, but the federal government finances the operation. In a purely federal deployment, troops can perform their duties anywhere in the world, but in a Title 32 mobilization, soldiers are limited to the continental United States, and foreign intelligence collection cannot be part of such an operation.

“Guardsmen in Title 32 status do not have the authority nor is that the intent,” said Colonel Davis, the Defense Department spokesman. “At this time there is no effort to update the policy,” he added.

In April, in rural Starr County in South Texas’ Rio Grande Valley, the soldiers stationed at two observation posts appeared to be following the guidelines.

The troops stood on rocky, muddy cliffs on the river’s edge, peering through binoculars and focusing most of their attention on the banks and the brush on the American side. The official boundary between the two countries at that point is not the border fence but the middle of the Rio Grande, and several of the troops at the outposts stood facing east and west, but never directly focused their binoculars south across the river into the brush in Mexico.

National Guard officials said the troops were carrying out their approved mission, and they did not view the surveillance rules as a prohibition or restriction but simply as part of the parameters of the deployment.

Their mission, National Guard officials said, is to monitor and detect, and to perform many of the administrative, logistical, maintenance and surveillance tasks that Border Patrol agents would be doing, so those agents can be freed up to be out in the field. Officials refer to the troops’ support role as being a “force multiplier.”