Saudi-Iranian political rivalry has aggravated tumult across the Middle East for years, but has escalated in recent months as Riyadh's new rulers have taken a harder line and as the nuclear deal has relieved pressure on Tehran.

Iran's international rehabilitation also opens the prospect of economic rivalry, with Saudi Arabia facing not only a fellow oil producer in an era of oversupply and low prices, but also a more self-reliant and multi-skilled economy.

Even without public pronouncements, Riyadh's private consternation could be discerned in the pages of semi-official media and comments by influential clerics.

The main cartoon in al-Watan daily simply showed a pencil broken mid-way through writing the word "peace", while an opinion piece underneath it asked "Will Iran change after the nuclear deal enters implementation?" Its answer: probably not.

Saudi Arabia, a conservative Sunni Muslim monarchy, sees revolutionary Iran as the paramount threat to the Middle East's stability, because of its support for Shi'ite militias that Riyadh says have inflamed sectarian violence and undermined Arab governments.

For the Al Saud dynasty, the nuclear deal was a double blow, freeing Iran from sanctions it believed helped check those regional activities and raising the specter of a rapprochement between Tehran and Riyadh's most powerful ally, the United States.

Under King Salman, Saudi Arabia has launched a war in Yemen to stop an ally of Tehran gaining power, mobilized Muslim states to freeze Tehran out of regional and Islamic influence, and has boosted support to rebels fighting Iranian allies in Syria.

Iran's pragmatist President Hassan Rouhani, who has publicly reached out to Riyadh repeatedly since his election in 2013, on Sunday rebuked the Al Saud for their own regional stance and called on them to "take the path that will benefit the region".

Such comments, which mirror accusations Riyadh makes about Iran, infuriate Saudi officials who regard Rouhani as a smooth-talking cipher for Supreme Leader Ayatollah Ali Khamenei and see no point engaging with him instead of his harder line superior.

They disdain the campaign by U.S. President Barack Obama to bring Iran in from the cold, believing him pusillanimous in the face of what they see as Iranian aggression and guileless in accommodating Iran's moderates when Khamenei pulls the strings.

TUMBLING STOCKS

Between the full-page adverts Saudi companies had run in Monday's newspapers to congratulate King Salman on the Islamic calendar anniversary of his becoming monarch, the opinion columns and cartoons despaired at Iran's comeback.

The role of Western powers, particularly Riyadh's oldest ally the United States, in facilitating Tehran's relief from pariah status and their hopes of cashing in on Iran's newly opened economy did not go unremarked.

The al-Jazirah daily's cartoonist showed an incarnation of American capitalism in striped trousers and top hat carrying a bottle marked "sanctions", from which emerged a genie in the guise of a Shi'ite militia fighter, his turban marked "Iran".

When Salman visited Washington in September, the main focus of talks was the push by his powerful son, Deputy Crown Prince Mohammed bin Salman, to win investment from American firms, an apparent riposte to Iran's overtures to Western companies.

Sunday's news that sanctions would indeed be lifted raised fears of a slide in oil prices below their existing low level of under $30 a barrel as Iran immediately ordered an increase in output, while Saudi stocks fell by 5 percent.

There could be no more pronounced contrast with the mood in Iran, freed from years of increasingly tough sanctions which have eroded its currency and allowed Saudi Arabia to eclipse its economy.

In 2000, Iran's gross domestic product was larger than Saudi Arabia's, according to International Monetary Fund data; now, the $650 billion Saudi economy is much bigger than Iran at $400 billion.

That trend may now start reversing, altering how far each country can afford to mount political and military adventures overseas, and the extent to which they can use trade relations to build alliances with foreign powers.

Saudi Arabia's economy is slowing sharply because of low oil prices, which the entry of new Iranian crude into the market will intensify. Iran is looking forward to a trade and investment boom as sanctions are lifted; it has big non-oil sectors such as agriculture and car manufacturing that the Saudis lack.

WIDER ALARM

The volatile nature of Saudi-Iranian relations, aggravated already this month by a diplomatic row following Riyadh's execution of a Shi'ite cleric, is causing alarm among world powers who fear things will get worse.

This week a procession of top officials from around the world will visit Riyadh, including Chinese President Xi Jinping, Pakistani Prime Minister Nawaz Sharif, French Foreign Minister Laurent Fabius and U.S. Secretary of State John Kerry.

While they all have other business to discuss, the question of Saudi Arabia's handling of its rivalry with Iran, and the wider risks it entails, are likely to be addressed, with both Xi and Sharif planning also to visit Tehran.

What they all fear is that if competition between the Middle East's foremost powers cannot be contained, it will complicate efforts to end wars and political struggles across the region or even break out into new fighting elsewhere.

Such concerns are only compounded by the sectarian lines along which the rivalry has become drawn, and the likelihood that hostility between Saudi Arabia and Iran will translate into religious competition that fuels militancy across the world.

A letter signed by 140 Saudi clerics, including prominent names, calling on the government to beware what they termed Iran's "record of criminality and treachery" and to support regional Sunnis cannot have assuaged international alarm.

Neither can the Tweet by Sheikh Saud al-Shuraim, imam of the Grand Mosque in Mecca, using a derogatory term often employed by Arab Sunnis to describe Iranians and Shi'ites, after the sanctions were lifted.

"There is no surprise in the alliance of the Safavids with the Jews and Christians against the Muslims, history witnessed this. But there is surprise at minds delaying their understanding of this truth until this moment," he wrote.

(Additional reporting by Andrew Torchia and Sam Wilkin in Dubai; Editing by William Maclean and Giles Elgood)