By Steve Holland

(Reuters) - In the hours after a gunman shot 12 police officers, killing five in Dallas, Republican presidential candidate Donald Trump and his Democratic rival Hillary Clinton abruptly canceled campaign events. Their Twitter accounts largely went quiet.

The shootings marked yet another convulsive event in the 2016 political season, one in which Clinton and Trump have scrambled to find the right response to terror attacks abroad, mass shootings at home, and protests over police killings of African Americans.

Much of the violence, captured live on smartphones and endlessly replayed on cable television news, has fueled Americans' fears about their personal safety, polls show. It has evoked memories of 1968, when civil rights leader Martin Luther King and Democratic presidential candidate Robert Kennedy were assassinated in a turbulent era of protests against the Vietnam war and racial segregation.

"We're seeing wild acts of gun violence and we are polarized in our politics and in the public square," said historian Douglas Brinkley. "Every 48 hours there seems to be some horrific event that jars and jerks our consciousness into a new dimension."

Historically, uncertain times tend to push the party out of power. Witness Republican Ronald Reagan's defeat of incumbent Democrat Jimmy Carter in 1980 during the Iran hostage crisis, or Obama's victory over Republican John McCain in 2008 as the U.S. economy collapsed.

But that may not be the case for Trump, who has yet to convince a majority of Americans that his would be a steady hand at the tiller.

Trump's initial reaction to Dallas suggested he understands the stakes. He issued a sober statement of unity that was unlike many of his public utterances that are often derided by critics as hyperbolic or inflammatory.

"Our nation has become too divided. Too many Americans feel like they’ve lost hope. Crime is harming too many citizens. Racial tensions have gotten worse, not better. This isn’t the American Dream we all want for our children," he said.

"This is a time, perhaps more than ever, for strong leadership, love and compassion. We will pull through these tragedies."

BRIDGING THE DIVIDE

With race relations becoming a more prominent issue in the White House race, former House Speaker Newt Gingrich, a top contender to be Trump's vice presidential running mate, told Fox News that Trump and his No. 2 can help broaden their appeal by going to inner cities, where many African Americans live.In responding to the spasm of gun violence, Clinton has called for more regulations on the gun industry and has aligned herself with the Black Lives Matter movement, which campaigns against violence against African-Americans.

In a round of TV appearances, Clinton said she had been working her whole life to try to bridge the racial divide. Her comments follow the police shooting of two black men in two separate incidents in Louisiana and Minnesota over the past week. The Dallas gunman told police he was angry about the killings.

“I will call for white people, like myself, to put ourselves in the shoes of those African American families ... who fear every time their children go somewhere. I'm going to be talking about white people,” Clinton said on CNN.

Democratic strategist Bud Jackson said she is hitting the right tone.

"I don't think this is an issue that she needs to take advantage of," he said. "In fact it would probably hurt her if she is perceived as taking advantage of it."

But the demand for change is sweeping among voters. According to a Reuters/Ipsos poll taken over the last five weeks, nearly two-thirds of Americans feel the country is on the wrong track, reflecting their general unease with the economy, terror threats and violence.

"Her challenge is to show she can cope with these issues and that she would bring change and would not be the same," said presidential historian Thomas Alan Schwartz.

Trump has said a Clinton win in the Nov. 8 election would amount to a third term for President Barack Obama. Clinton has embraced many of the policies of her former rival in the 2008 presidential election and campaigned with him this week.

Lanhee Chen, who advised 2012 Republican nominee Mitt Romney and 2016 presidential candidate Marco Rubio, said a more civil discourse in American politics would help to calm people's general fears.

"I think there's bigger issue around civility and rhetoric and discourse and how our leaders appeal or don’t appeal to our better angels. That’s really what this is about. I see this as an absence of leadership all around."

(Reporting By Steve Holland; Editing by Paul Thomasch and Ross Colvin)