* Bahrain seeks military help despite U.S. urging dialogue
* Risks seen for Washington, opportunity for Tehran
* Oil jitters complicate tricky situation
(Repeating analysis moved on Tuesday)
By Andrew Quinn
WASHINGTON, March 15 (Reuters) - Saudi Arabia's military intervention in Bahrain has exposed a diplomatic rift as Riyadh and Washington make different calculations over a crisis that could have a far-ranging impact on their relations.
Close Saudi-U.S. ties anchor stability in the oil-rich Gulf. The autocratic Sunni Muslim monarchy provides 12 percent of U.S. crude imports and serves as a powerful regional counterweight to Shi'ite-ruled Iran, a defiant U.S. foe.
But when Saudi troops rolled across the causeway into neighboring Bahrain on Monday to buttress the embattled Sunni royal family against protests by the island's Shi'ite Muslim majority, Washington scrambled to respond as its own repeated pleas for negotiation appeared to have been swept aside.
"This is the tightrope the administration has to walk," said Kenneth Pollack, a Middle East expert at the Brookings Institution, a Washington think tank.
"On the one hand they very much do want to embrace these kind of demands for democratic change. On the other hand, they do not want to wind up on the opposite side of any kind of confrontation with the Saudis," he said.
The White House reacted carefully in giving the Obama administration's view of the Saudi military move, insisting it was not an "invasion", while urging Saudi Arabia and other Gulf states deploying forces to Bahrain to show restraint.
Few suggest that the fundamentals of the U.S.-Saudi relationship -- built on oil, counterterrorism cooperation and a shared wariness of Iran -- are under threat.
But Saudi Arabia's involvement in Bahrain injects an unpredictable new element into the mix that could backfire if the tiny kingdom's political violence escalates.
"The long-standing pillars of the relationship are still solid," said Frederic Wehrey, a senior policy analyst at the Rand Corporation.
"But if Bahrain really goes south, there is a serious outbreak of violence and perhaps an overreaction by Saudi security forces, I think it could quickly escalate into a major issue between the two sides."
White House spokesman Jay Carney said on Tuesday the United States was aware of Bahrain's request for help from its neighbors and regarded Saudi Arabia as "an important partner."
Bahrain's crisis comes as Washington struggles to respond to protests across the Middle East that have toppled U.S.-allied governments in Egypt and Tunisia, rocked another in Yemen, and sparked a bloody rebellion against Libyan strongman Muammar Gaddafi.
The Saudi move into Bahrain raises uncertainties over two key U.S. interests: the price of oil and the role of Iran, which is already locked in a dispute with major western powers over its nuclear program and is seen by Washington as a dangerous troublemaker in the region.
Climbing oil prices, now at around $100 per barrel, have raised fears for the fragile U.S. economic recovery, and officials say potential further supply disruptions in the Gulf or elsewhere could exacerbate the situation.
Saudi Arabia has pumped oil to make up for supplies lost due to Libya's turmoil. But its main oil-producing regions lie near Bahrain in the east, where existing sectarian tensions between Sunni and Shi'ite Muslims could be inflamed by any surge in violence across the border.
Iran called the dispatch of Saudi troops to Bahrain unacceptable and political analysts say the dispute could widen with unknown consequences.
Toby Jones, a Saudi Arabia expert at Rutgers University, said Riyadh may be flexing its muscle both for Tehran and for Washington -- to demonstrate that it will act when it sees its own interests at risk.
"The Saudis think they have some leverage over the U.S., which gives them a freer hand," he said. "It's a gamble."
Saudi Arabia took matters into its own hands after the region's rulers watched the United States drop long-time allies such as Egypt's former President Hosni Mubarak to support pro-democracy protests -- a move U.S. President Barack Obama said put Washington "on the right side of history."
"It reflects a concern about how steadfast is America's commitment to their survival," said Robert Danin, a Middle East expert at the Council on Foreign Relations.
Both U.S. Secretary of State Hillary Clinton and Defense Secretary Robert Gates dropped plans to visit Saudi Arabia this month due to the king's health. But Clinton met the Saudi foreign minister in Cairo on Tuesday and told him the United States hoped for a peaceful resolution in Bahrain.
"Our advice to all sides is that they take steps now to negotiate toward a political resolution," she said, adding the security situation was "obviously important".
Senator John Kerry, the Democrat who chairs the Senate Foreign Relations Committee, said the United States was still on the same page as its Saudi allies.
"I understand what they are doing. I think they are pushing for a dialogue. They're looking for reform. It is not an occupation," he told Reuters. (Additional reporting by David Alexander, Arshad Mohammed Mark Hosenball and Susan Cornwell; editing by John Whitesides)