President Mahmoud Ahmadinejad will make history when he arrives in Baghdad on Sunday, the first Iranian president to step foot in Iraq on a trip many see as symbolic of Tehran's new influence over its old foe.

Iranian sway over Baghdad has grown substantially since the 2003 U.S.-led invasion ousted Tehran's long-time enemy Saddam Hussein and his ruling Sunni Baath party against whom it fought an eight-year war in the 1980s.

But just how far this influence extends is far from clear.

Some experts argue that Tehran wants to dominate its neighbor, wielding political, economic and religious control. Others say it neither desires nor is able to achieve such ends.

The evidence is inconclusive and murky, mirroring the often opaque nature of Iranian politics itself.

Iran has the ability to destabilize the political situation in Iraq by triggering violence, by supporting and arming some militia and through its influence on the Iraqi government, said Hasan al-Shimmari, head of Iraq's small Shi'ite Fadhila Party.

The political system in Iraq is not mature enough yet. It gives a chance for Iran to interfere in Iraqi matters.

U.S. officials say Iran exerts considerable influence over Shi'ite cleric Moqtada al-Sadr, who has a large following among Iraq's Shi'ite poor and commands a powerful militia, the Mehdi Army. Tehran is widely believed to have been behind ceasefires called by Sadr in August and again this month.

The respected London-based Chatham House think-tank said in a report in 2006 that Tehran now had more influence over its neighbor than Washington, not only in government circles but at street level as well.

Such reasoning is based on long-established social ties between the Shi'ite parties that dominate the government in Baghdad and predominantly Shi'ite Iran, where the powerful Supreme Islamic Iraqi Party and other Shi'ite parties had bases during Saddam's regime.

That has given Iran significant clout, particularly in the Shi'ite-dominated south, and the ability to influence decision-making, according to Vali Nasr, a Middle East expert at the Council on Foreign Relations in Washington.

Iran wants to protect its national interest in Iraq, which means a friendly, pliable Iraq: no return of Baathists to power, an Iraq controlled by Shi'ites open to business, trade and influence from Iran, he told Reuters.

But others disagree. I have seen no sign that Iran wants to shape Iraq politically, Peter Harling, a Damascus-based analyst with the International Crisis Group, told Reuters.

The impression we get is more that Iran is hedging its bets so it can react to any scenario as it unravels rather than impress its own agenda on Iraq.

NO COHERENT POLICY?

So, why are there such diverse opinions? Dr Rosemary Hollis a senior researcher for Chatham House says part of the problem is that the Iranians themselves have no coherent policy.

Some in Tehran want a stable Iraq that will benefit them politically and economically. Others are more concerned about U.S. forces leaving Iraq, beaten and humiliated.

That has been the dichotomy of their policy, said Baqer Moin, a London-based Iranian analyst.

Washington has accused Tehran of waging a proxy war in Iraq by training and arming Shi'ite militias, particularly rogue elements of Sadr's Mehdi Army, that it seeks to use to keep the Baghdad government weak, a charge denied by Iran.

Nasr says Tehran brokered a deal between Sadr, who is thought to spend much of his time in Iran, and his main Shi'ite rival, the Supreme Islamic Iraqi Council, to prevent a conflict between the two groups in the southern oil hub of Basra.

But long-standing links do not mean Iran can control such groups and nor can they rely on Tehran.

Analysts said the rivalry between Shi'ite factions meant Iran had to tread a cautious line when dealing with them, and this limited how much it could control each group directly.

The mistake that the Iranians always make is they assume that having long-standing ties and doing a favor translates into people taking orders, said a Western diplomat in Tehran.

There are certainly good economic reasons for Iran to want to hold sway, particularly in the south of the country where much of Iraq's oil reserves are located and where Shi'ite groups are jockeying for control.

The (Iranians) have been rushing to help with development projects not just arming militia to oust the Americans, Chatham House's Hollis said.

Despite the fears that the United States and many Sunni Arab states in the Middle East have about Iran's intentions for Iraq, experts believe it is one issue on which Washington and Tehran can find common ground.

Iran and the U.S. have all along supported the same government and politicians in Iraq, Nasr said. They have more shared interest in Iraq than in any other issue in the region.

Iraq