For example, Ayotollah Sistani was not even considered by the Americans prior to the invasion because he held no sway over Saddam's policies.  But once Saddam was removed, Sistani was the most powerful man in Iraq.  Sistani has been mostly a moderating force in Iraq since the invasion, but his version of Islam is much less compatible with Sunni society than the secularized cosmopolitan version that prevailed in Baghdad prior to the war.  Sunnis rebelled against the very idea of Sistani having sway.  

This isn't to say that there is no Sunni-Shiite tension. There is, of course. Saudi Arabia is a prime example of where the Shiite minority is oppressed.

But they are not natural enemies by any means. That is why Hizbullah's Hassan Nasrallah, along with Ahmadinejad are the 2 most popular politicians in, for example, Egypt. And if the Muslim Brotherhood were ever to overthrow Mubarak, Iran would be their first choice for an ally.

During that 1400-1500 year period there have been only three significant incidents of what could be termed sectarian violence between Sunnis and Shi`as. All three of those incidents were a result of a foreign invasion and occupation. In each case once the foreign occupation ended, relations returned quickly to normal.

I would be happy to provide more details later if anyone is interested, but that's it in a nutshell.

"A decision to introduce into the Azhar's curricula the study of Zaidite and Twelver Shi'ah (Ja'farite or Imamite) fiqh on a par with the fiqh of the four Sunni schools (madhhabs). This last step is one which the new Rector had long ago advocated in his earlier writings; according to the Azhar Journal, [5] he and Shaikh Mustafa al-Maraghi frequently adopted the Shi'ah view rather than any of the Sunni views on disputed points of fiqh. A Sunni-Shi'ah Rapprochement Society (Jama'at al-Taqrib) was founded as long ago as 1948, on the initiative, it is said, of an Iranian Shi'ah named Shaikh Muhammad Taqi Qumi; and this society, under the leadership of an Iranian named Shaikh Kazim, continued to enjoy the Azhar's sponsorship."

(snip)

"Firstly, Islam does not oblige any of its followers to follow a particular madhhab but grants to every Muslim the right to adhere initially to any madhhab which has been correctly transmitted and whose precepts are documented in specific books, and that the adherent of any of these madhhabs may shift to another one without there being any objection to his so doing; and

Secondly, that the madhhab known as the Twelver Shi'ah Imamite is one in which worship (ta'abbud) may be legally performed on a par with the madhhabs of Sunnis." "Muslims should know this and refrain from fanaticism in favour of a particular madhhab. God's religion and law do not belong and are not confined to any one rnadhhab. They are all earnest seekers for truth (mujtahidun), rewarded and accepted by God. It is permissible that anyone not qualified to have insight or to search for truth (ijtihad) should follow one of them (taqlid) and carry out the prescriptions of its fiqh alike in worship and in conduct ('ibadat wa mu'amalat)."

You also appear to be conflating sectarian conflict with political conflict and government suppression of opposition. Admittedly, they sometimes appear to be pretty tightly entangled, but in my experience the basis for a conflict is usually either politically or religiously based, and less usually both.

"You can see evidence of that even in Baghdad, where Saddam City (now Sadr City) was a large slum for Shi'ites and where resistance to the regime was strongest."

Maybe I am putting too fine a point on it, but what is now known as Sadr City (originally named Madinat Al Thawra - Revolution City) was never "for Shi'ites". It was built in the late '50's, early '60's by Abdul Karim Qasim to provide low cost housing. By coincidence, not by design, the majority of its early population were migrants from rural Maysan province and were mostly Shi`as. Sunnis, Christians, and Jews also lived there. There may still be some Christians there - don't know -  but there were Christians living there at the time of the 2003 invasion and there are still churches there. Somewhere in the back of my mind I have an inkling that there was a synagogue there, but it has been a long time, so that could be a figment of my imagination.

The original basis of the resistance in Sadr City was  not so much Shi`a versus Sunni, but more religious state versus secular state. Perhaps that is another discussion for another time.  

"You could see it in Saddam's refusal to allow annual Shi'ite pilgrimages to Najaf and Kufa and Kerbala."

The reason for that was not sectarian based. The reason for that was that the strongest and most open threat to the regime was from parties, led mainly by Shi`a authorities, that aimed to overthrow the secular regime and turn Iraq into an Islamic state. In fact, the Da`wa party, one of the main opposition parties, was not strictly Shi`a, and its goal was to work with Sunni leaders to transform Iraq into an Islamic state. Saddam stopped the pilgrimages not for sectarian reasons, but to protect the regime from the threat of  huge gatherings of emotionally hyped-up people whose leaders wanted to overthrow it and impose their own form of religious rule.