* Putin watches sport, meets other teams at Games
* Projects softer image than usual to the world
* Putin avoids controversy at Games, builds bridges
By Timothy Heritage
SOCHI, Russia, Feb 16 (Reuters) - Vladimir Putin says sport and politics do not mix but his actions at the Winter Olympics suggest the opposite.
When he was not watching ice hockey in Sochi at the weekend, the Russian president was doing the diplomatic rounds, dropping in on foreign teams, sipping wine or tea with them, talking sport - and building bridges.
He also found time to visit an injured Russian athlete in hospital and press the flesh with fans at the ski venues in the mountains above Sochi.
The message was that more than a week after he declared the Games open, these are still very much Putin's Olympics, a global event on which his legacy depends.
Back in Sochi after a four-day interval, his boldest step was to visit the U.S. team on Friday and sip red wine on the sofa with American officials.
"What I like in you is that you have strong competitors in almost every discipline, in almost every event you fight for medals," he said in footage that led the evening television news in Russia and was beamed around the world.
Putin looked at ease with the Americans, despite President Barack Obama's decision not to come to Sochi and months of criticism by U.S. gay rights activists of what they see as Russia's persecution of homosexuals.
The meeting amounted to the extension of an olive branch to a country with which Putin is at odds over Syria's civil war, human rights and democracy, and Russia's decision last year to grant asylum to ex-U.S. spy agency contractor Edward Snowden.
When an American referee disallowed a goal that might have sealed a famous victory for Russia against the U.S. ice hockey team on Saturday, Putin refused to criticise the decision.
"Sport is sport," he said, despite looking upset.
Putin later went to see the Ukrainian team, a sensitive visit because of anti-government protests in Kiev provoked by a decision to forego a trade pact with the European Union and rebuild economic ties with Moscow.
The president regards Ukrainians as Russia's Slav brothers but the protesters fear he wants to suck their country back into Russia's orbit.
There was no mention of the unrest in the Ukrainian capital or Russia's delay in handing over a loan it has promised Kiev. Putin was all smiles again and he carefully dodged a tricky question in the interests of diplomacy.
When one of the Ukrainians criticised the American referee against Russia, Putin made clear those were not his words but laughed: "Okay. It's not very nice to hear that from you."
It was only at another meeting, with the Swiss team, that Putin briefly let down his guard - but to a reporter.
Asked about the prospects for political liberalisation after the Olympics, he replied testily: "Is there any hope you will not link sports with politics?"
There was no hint of irony, and it was then back to the softer image Putin is projecting at the Games.
Up on the mountain slopes above Sochi on Sunday, he stopped to pose for a photo with a surprised volunteer worker at the Games.
On Saturday evening, he visited Russian skicross racer Maria Komissarova in the hospital where she underwent surgery for a broken back, sustained in training. He even called her father when she told him he was worried about her.
This is the "new" Putin on show at the Olympics. The pictures of him in action - flying with cranes, riding horses bare-chested or shooting a tiger with a tranquiliser dart - have at least for now been carefully put to one side.
The Russia that Putin wants to portray at the Games is a caring country that has come a long way since the austere days of the Soviet Union. The leader he wants to portray is a man with whom the West can do business.
Even Russian opponents seem to have accepted that now is not the time to criticise him, while the rest of the world wants a peaceful Olympics. There have been few protests during the Games and his main critics have been quieter than usual.
For now he can bask in the glory, hope Russians' outrage over the disallowed goal pipes down and the Russians bring gold in the ice hockey on the last day of the Games.
But if they fail, do not expect him to criticise the referee.
(Additional reporting by Alexei Anishchuk; editing by Keith Weir)