By Mary Milliken and Steve Keating

LOS ANGELES/RIO DE JANEIRO (Reuters) - America's hot summer of racial tension, violence and politics has spilled on to the basketball court, the track and the Twitter feeds of some of the country's top sports stars.

Now, as athletes head to the Olympic Games in Rio de Janeiro from Aug. 5-21, will they take the debate over America's racial divide onto the world's biggest global sports stage with them?

What comes to mind is another summer in which sports and race collided spectacularly: in 1968 at the Olympics in Mexico City.

There, two American 200 meters track medalists, Tommie Smith and John Carlos, raised their black-gloved fists and bowed their heads on the podium in support of Black Power, a protest for which Carlos says he has paid dearly.

"As soon as we raised our hands, it's like somebody hit a switch. The mood in the stadium went straight to venom," Carlos wrote in an op-ed on Vox in July.

"Within days, Tommie and I were suspended from the US Olympic team and had to leave Mexico City early."

The violence on America's streets has made many think back to the 1960s, the decade defined by the fight for civil rights.

In July alone, two black men were killed by police in Louisiana and Minnesota, and there were two ambushes on police in Dallas and Baton Rouge, Louisiana that left eight officers dead.

Some athletes, like women in the WNBA pro basketball league, have shown support for the Black Lives Matter movement that coalesced nationally in 2014 after the police killing of an unarmed teenager in Missouri.

Others like basketball player Carmelo Anthony, a member of the Olympic team, have used their star power to bring together communities and police to discuss racial tension and policing.

"I am so humbled and honored that the women of the WNBA went to such an extent to uplift our message and the cause," said Kofi Ademola, lead organizer for Black Lives Matter in Chicago.

Ademola said he is watching to see if Anthony and other athletes follow the "powerful women from the WNBA" or support the police like some entertainers have or "remain silent and oblivious to everything, as if nothing is going on".

The WNBA initially fined teams for wearing Black Lives Matter t-shirts or black warmups last month, but then backtracked after widespread outrage. Tina Charles, who is on the Olympic team, tweeted "I refuse to be silent @WNBA" with a picture of her in a t-shirt emblazoned with #BlackLivesMatter and #Dallas5.

"Appreciate our players expressing themselves on matters important to them," WNBA President Lisa Borders tweeted after she reversed course. "Rescinding imposed fines to show them even more support."

ABIDING BY RULE 50

Other athletes have chosen to make more neutral statements about peace and pride in their country.

After winning the 200 meters final in the U.S. Olympic track trials days after the Dallas police killings, Justin Gatlin urged people to take the love in the stadium "and give it to someone you have never loved before. Just go up to them and say I love you for being an American."

In the violence-filled first week of July, Crystal Dunn, a member of the U.S. women's soccer team, tweeted "ANY wrongful death should concern everyone," and noted that she has been getting "a lot of questions lately about how I feel about being a black player on the national team."

"Although I am Black, I am a human first...," she added.

Public opinion appears to be against using the Olympic stage to make statements.

Around two-thirds of Americans want Olympic athletes to keep their political views to themselves in Rio de Janeiro, according to a Reuters/Ipsos poll released on Monday.

When asked how the U.S. Olympic Committee would deal with athletes who made statements or protested, a spokesman pointed simply to the Olympic Charter's Rule 50.

That rule discourages participants from using the Games as a platform for protests or "the promotion of political, religious or racial propaganda".

And while the International Olympic Committee says athletes are free to express their opinions, they are advised to only wear official uniforms on the field and at ceremonies.

The place to make a statement might be at one of the Games' last events, the men's basketball gold medal game on Aug. 21. If protesters get sent home, they will not miss much at all.

(Additional reporting by Amy Tennery, Scott Malone, Liana Baker and Gene Cherry. Editing by Ed Osmond)