What does “go rogue” mean in “Donald Trump goes rogue left and right”? Is “go rogue” a common idiom?

There was the following passage in the New York Times' article that came under the headline, “Sarah Palin endorses Donald Trump, which could bolster him in Iowa”:

“As Mrs. Palin announced her backing, Mr. Trump stood wearing a satisfied smile as she scolded mainstream Republicans as sellouts and praised how Mr. Trump had shaken up the party. “He’s been going rogue left and right,” Mrs. Palin said of Mr. Trump, using one of her signature phrases. “That’s why he’s doing so well. He’s been able to tear the veil off this idea of the system.”

As I checked the meaning of “go rogue” with both Oxford and Cambridge online dictionaries, neither of them carries “go rogue” as an idiom. Whilst Urban dictionary carries the definition of “go rogue” as “to cease to follow orders; to act on one's own, usually against expectation or instruction. To pursue one's own interests.”

If I follow Urban Dictionary’s definition, does “Trump has been going left and right” mean he has kept going against expectations of the left and the right?

What does “He’s been going rogue left and right,” mean, and why “go rogue” is Mrs. Palin’s signature phrase? Is it so creative, impressive, and effective phrase as being called as a signature phrase?

"Going rogue" has a peculiar resonance with a political party (like the U.S. Republican Party) whose symbolic animal is an elephant—namely, the historical connection between "going rogue" and "rogue elephant." Here is the definition of rogue elephant in Merriam-Webster's Eleventh Collegiate Dictionary (2003):

rogue elephant n (1859) 1: a vicious elephant that separates from the herd and roams alone 2: one whose behavior resembles that of a rogue elephant in being aberrant or independent

Roger Courtney, Africa Calling: The True Account of the Author's Strange Workaday Experiences in Kenya, Uganda, and the Belgian Congo (1935) [combined snippets] describes the phenomenon of elephants going rogue:

Often enough a lone animal will go 'rogue.' One reason for going 'rogue' is tuskache—a super-toothache. I remember one such animal creating a reign of terror over a whole district, smashing senselessly through shambas, destroying right and left. Woe betide any person or thing that stood in his path. At any time of the day or night there would come to the terrified natives a sudden squealing and high trumpeting, and the maddened monster would come thundering upon them out of the bush. The hunters employed to hunt down this dangerous beast frequently came across holes where the animal had thrust his aching tusk into the ground. Elephant often dig in the ground with their tusks, searching for roots, but the frequency with which this one did so, also the fact that it was damp, cool soil which he favoured, left no doubt that he was seeking to relieve the agony in his tusk. When eventually he was run to earth it was found that, because of some injury sustained long before, one of his tusks was badly decayed high up.

The idiom "go rogue" is fairly recent in U.S. English. For much of its career, it had a rather unsavory connotation, as this brief parenthetical from SSCP Systems Security Certified Practitioner Study Guide and DVD Training System (2003) indicates:

Documentation, while often boring, serves a very critical purpose. One of the criteria most commonly found in security audits relates to the existence of documentation. If the administrator who set up all of the security were to leave the company or go rogue (become malicious), someone else will need to take over their job.

And from James Stefanie, The Charters Affair: Being a Reminiscence of Dr. John H. Watson (2000):

"Unfortunately, not every part of the body functions properly and in good relation to the other parts all the time. A cell may go rogue and infest the blood with toxins; another cell may grow madly and tumors sprout. As a doctor I am charged with taking the whole of the healthy body and protecting itself from one of its members gone wrong. Does this mean to me that the whole body is evil because one part becomes malevolent? Or does it mean I, as a physician, must keep the body from becoming victim to itself?

But at some point in the past eight years, "going rogue" ceased to be exclusively the province of elephants with aching tusks, and malicious computer security administrators, and toxic body cells, and (perhaps) U.S. Army colonels overseas who have left the (military) reservation in troubling ways. It has become in its new sense, as Rathony's answer observes, a badge of honor—an indiscriminate rampaging with a higher purpose, as it were.

Sarah Palin's book Going Rogue: An American Life (2009) was surely the turning point in the adoption by a segment of U.S. society of the idea that going rogue might be a good thing in a society where, presumably, everything likely to get stepped on is rotten and deserves to be trampled. Hence the new "beholden-to-no-one maverick" sense of "going rogue" that Rathony mentions.

Would Sarah Palin and Donald Trump benefit from pain-free dentistry? I doubt it. In any case, for the time being, the idiom "go rogue" has two highly incompatible meaning: on the one hand, "go wild and put a lot of innocent people at risk of harm"; and on the other, "escape the control of a bunch of cynical handlers and moneyed elites who normally suppress the truth, control the terms of public discourse, and promote the status quo."

The "left and right" part of the quotation, I believe, is not meant to indicate that Trump is veering leftward and rightward (politically speaking) in unexpected ways, but that he is pressing the attack in all directions. Idiomatically, "doing something left and right" simply means doing something vigorously or in all directions, such as "knocking them down left and right."

It is no more an idiom than "go crazy," "go apeshit," or "go to town" are idioms.

Go rogue in this case is the exact opposite of toe the party line.

What Palin meant was that Trump is in the habit of saying things that are not in keeping with the Republican Party's doctrine, set or principles, ideology, etc, etc. The unintentional ambiguity ("left and right") is in keeping with Palin's elocution. ("Left and right" does mean "at every turn"; however, it could also mean "leaning now towards the liberals, now towards the conservatives." Such such subtleties seem to be beyond her, though).

Adding to Ricky's great answer, North Korea is often times referred to as a rogue nation by the media because (1) it is completely segregated from the international community, (2) you never know how it will act or respond as it is very difficult to predict its behavior including nuclear bomb tests and (3) it is very difficult to control.

The word rogue was used to describe large wild beast living apart from the herd from 1859, originally of elephants.

1560s, "idle vagrant," perhaps a shortened form of roger (with a hard -g-), thieves' slang for a begging vagabond who pretends to be a poor scholar from Oxford or Cambridge, which is perhaps an agent noun in English from Latin rogare "to ask."...

In playful or affectionate use, "one who is mischievous," 1590s. Meaning "large wild beast living apart from the herd" is from 1859, originally of elephants. Meaning "something uncontrolled or undisciplined" is from 1964. Also common in 17c. as a verb. Rogue's gallery "police collection of mug shots" is attested from 1859.

Rogue can have a positive and negative connotation at the same time. If it is used for North Korea, it is negative, however, if it is used for a politician like Donald Trump by someone who supports him, it is positive and its meaning is closer to political maverick which is used to mean:

a lone dissenter who takes an independent stand apart from his or her associates

If you compare the etymology of the two words, you will notice they have something in common, i.e., elephant apart from the heard and unbranded calf separated from its mother.

"He’s been going rogue left and right" means he (Donald Trump) has shown more independent and flexible stand compared with other Republican candidates who are trying to follow the core principles of the Republican Party.

[Online Etymology Dictionary, Dictionary.com]