DALLAS — The questions being raised by right-wing bloggers and conservative commentators about a Pentagon training exercise called Jade Helm 15 traverse the outer edges of political paranoia.

The eight-week exercise starting in July and planned for locations in Texas, New Mexico, California and other Southwestern states, they say, is part of a secret plan to impose martial law, take away people’s guns, arrest political undesirables, launch an Obama-led hostile takeover of red-state Texas, or do some combination thereof.

But when Gov. Greg Abbott issued a directive to the Texas State Guard to keep watch over the military operation, seeming to give those concerns some official credence, it raised a more credible question: To what degree has the extreme become mainstream in Texas?

In his directive, issued last week to Maj. Gen. Gerald Betty, commander of the Guard, Mr. Abbott wrote that he wanted Jade Helm 15 monitored because it was “important that Texans know their safety, constitutional rights, private property rights and civil liberties will not be infringed.”