By Steve Holland

MIAMI (Reuters) - Donald Trump was in the middle of creating another nickname for Democratic rival Hillary Clinton - "Hillary Rotten Clinton!" - when he suddenly thought of his mild-mannered vice presidential running mate, Mike Pence.

"You think Mike Pence would say this?" he smiled to the audience in Roanoke, Virginia. "What a high-quality person."

On their first campaign swing together this week, Trump and Pence are giving voters an initial look at their odd-couple pairing as they prepare to battle against Clinton and her running mate, Tim Kaine, the Virginia governor, in the Nov. 8 presidential election.

Their public routine is a work in progress, but it's clear that they've learned from their first joint appearance, a sometimes awkward interview on CBS's "60 Minutes."

In that session, Trump frequently talked over Pence and Pence went out of his way to wave off his differences with Trump on policies like his past support for the North American Free Trade Agreement, which Trump wants to renegotiate.

They now have a system: When they campaign together, Pence introduces Trump enthusiastically as the modern-day equivalent of 1980s Republican President Ronald Reagan and speaks of himself as the son of a combat veteran and the father of a U.S. Marine. Then he steps aside.

The Trump-Pence union was a political marriage of necessity. The combustible, free-wheeling Trump needed the staid, staunchly conservative Pence, the governor of Indiana, to give him some credibility with establishment Republicans and a boost in the Midwest.

The risks for Pence seem minimal and the upsides are huge. He opted out of the 2016 Republican presidential sweepstakes in the fallout over his fight over a religious freedom law in Indiana that critics said was anti-gay, and he faced a potentially difficult re-election race for governor.

With his experience as a former member of Congress, he could serve as a conduit to lawmakers in a Trump administration, something that could be important for a presidential candidate who has never held elective office.

In the two weeks since he was named Trump's running mate, Pence has become a lightning rod for criticism from Democrats who portray him as holding extreme right-wing views.

Clinton's campaign chairman, John Podesta, described Pence as "an incredibly divisive and unpopular running mate known for supporting discriminatory politics and failed economic policies that favor millionaires and corporations over working families."

WHEN THE CAMERAS ARE OFF

Pence seems to be still getting accustomed to Trump's worldview. While introducing Trump in Roanoke on Monday, he hewed closely to the establishment Republican position in criticizing Clinton for not doing more to ensure that a contingent of U.S. troops was left in Iraq after a U.S.-Iraqi agreement expired.

"Remember, it was Hillary Clinton who squandered the gains that were made at the close of our presence in Iraq, the failure in judgment that sent ISIS on the loose," Pence said in Roanoke, using an acronym for the Islamic State militant group.

That was at odds with Trump's view that the American invasion of Iraq was wrong. "I didn't want to go to Iraq," Trump said moments after Pence spoke.

Still, Pence's loyalty to Trump seems absolute.

In his speech introducing Trump at their appearance before the Veterans of Foreign Wars convention in Charlotte, North Carolina, on Tuesday, Pence tried to show what Trump is like behind all the bluster.

"I know this man's heart," Pence said. "I know the way he speaks, when the cameras are off."

Trump chose Pence over former House of Representatives Speaker Newt Gingrich and New Jersey Governor Chris Christie, the two other finalists.

Although Republican elites have been divided over Trump's presidential bid, many approve his choice in a No. 2.

"Mike Pence is a good man. He will add value to the ticket," tweeted Jeb Bush, the former Florida governor and one of Trump's former rivals for the nomination.

Gingrich said via email that Pence is proving to be "a good balance for Trump's powerful personality."

Trump himself jokingly brought up the vetting process that led him to Pence, describing a candidate who was squeaky clean compared to others whose background checks turned up "many pages" of materials.

"With Mike, they came back with like nothing. He's the most perfect human being, this guy. I love him. Where is he? I love him," Trump said.

(Editing by Leslie Adler)