(CNN) The gulf between those who have occupied the Oval Office and those who crave it is on rare display Tuesday.

On the campaign trail, Donald Trump and Hillary Clinton are waging the combative politics that has defined a bitter, divisive campaign as they gear up for a crucial three-week period of conventions and vice presidential picks.

Trump told the Associated Press Monday that the violence of the past week "might just be the beginning for this summer" and will hold a rally Tuesday with Mike Pence, the Indiana governor who is a top potential vice presidential pick and could help Trump secure the trust of social conservatives. Clinton, meanwhile, held a rally Tuesday with Bernie Sanders to shore up the progressive vote to take on Trump.

But in Dallas, the mood was far more somber as President Barack Obama and his predecessor, George W. Bush, put their own partisan differences aside and gathered for a memorial service for five police officers slain last week. Obama spent much of Monday out of public view preparing his remarks, and his appearance with Bush offered a rare moment of unity and bipartisanship to a country struggling to bridge racial and ideological divides.

"We are not as divided as we seem," Obama said. "I know that because I know America."

Searing tests

Taken together, Tuesday's flurry of events represent tests for the country's leaders -- past and present -- amid a surge of public anxiety over terror attacks, gun violence, racial tensions and economic dislocation that is shaping the presidential race and threatening to deliver a late-term blow to Obama's legacy.

The rituals of eulogizing the dead and consoling grief stricken relatives that Obama will endure Tuesday were painfully familiar, after a string of visits during his presidency to cities scarred by mass shootings.

"I have spoken at too many memorials during this presidency," Obama said.

He faced an extreme balancing act during the service .

He sought to honor police officers gunned down while on duty by an African-American man who said he wanted to kill white cops. But Obama also worked to give voice to the anger and fear over a spate of killings of black men by police that has triggered deep anger and fear in minority communities.

Obama's challenging task was to paint some national narrative of healing that could ease tensions enough to allow frank discussions on an issue that threatens to spiral out of control and claim more lives in a senseless escalation of violence.

"He has got to acknowledge the lethal and ferocious assault on that police department there in Dallas," said Michael Eric Dyson, author of the book, "The Black Presidency: Barack Obama and the Politics of Race in America."

"And he has got to acknowledge the lethal and ferocious assault on black men and women's lives in America," Dyson told CNN's "Wolf" show.

Obama has walked that rhetorical tightrope as he weighed in on killings of black men in Louisiana and Minnesota.

"When people say 'Black Lives Matter,' that doesn't mean blue lives don't matter," Obama said in Poland. But within hours, the depth of the problem was revealed when five "blue lives" -- those of the Dallas policemen -- were taken in Dallas, a city that ironically had been earning plaudits for improving relations between its Police Department and minority communities.

Obama's former personal aide Reggie Love told CNN Monday he doesn't know how Obama balances the situation, as well the President's own internal conflicts.

"He's definitely the first African-American president so this is something that's always heavy for him, but he's also the leader of the free world," Love told CNN's "New Day." "I think it's important to unify the country as a whole and not just address one specific gender or one specific race."

Show of unity

In a show of unity, Bush made a rare appearance on the national stage alongside his successor -- to help salve the grief of a city where he has made his post-White House home and built a presidential library since leaving office.

His presence was a reminder that he will be conspicuous by his absence at another event where a former President would have been expected to play a visible role -- the Republican National Convention next week in Cleveland.

Bush, his father former President George H.W. Bush and his brother, former GOP candidate Jeb Bush, are among hordes of Republican establishment figures who will skip the quadrennial GOP party because of their discomfort with Trump.

The presidential race, meanwhile, will rage on.

Clinton and Sanders held a joint rally in Portsmouth, New Hampshire -- a moment of Democratic Party unity that often looked like it would never happen.

The primary rivals began the work of folding Sanders' "political revolution" into Clinton's establishment campaign, following exhaustive talks on the party platform which offered Sanders concessions on issues including health care and education.

The Vermont senator offered Clinton only tepid credit for her victorious primary campaign after they slogged through bitter primary and caucus battles, so the body language on Tuesday was important -- and sometimes awkward.

Trump has his own political divides to bridge on Tuesday -- and it is possible that his appearance with Pence could be a taste of things to come if he chooses the longtime conservative favorite as his vice presidential nominee.

Leslie Lenkowsky, a Hoosier State politics expert at Indiana University, said Pence's credibility among social conservatives could counter the presumptive Republican nominee's weakness with the crucial GOP constituency -- especially if he makes the GOP ticket.

"He certainly brings absolutely solid conservative credentials -- Trump has been criticized by Ted Cruz and some others for not really being conservative," he said.

But it is Trump who will set the tone Tuesday.