By Maha El Dahan and Edmund Blair
CAIRO, Dec 21 (Reuters) - Egyptians returned to the polls on Wednesday in a staggered parliamentary election after five days of violence in Cairo that has cast a pall over the transition to democracy and drawn a U.S. rebuke of Egypt's security forces.
Tahrir Square and surrounding streets were quiet through the night for the first time in a week. A night earlier, police and soldiers had used tear gas and batons to chase protesters demanding an end to army rule out of the square.
The latest confrontations, in which 13 people have been killed, made for a turbulent backdrop to Egypt's first election since Hosni Mubarak was toppled in February. Even before the vote got under way in November, a flare-up in Tahrir killed 42.
Nine provinces, mostly outside the capital, were holding run-off votes on Wednesday and Thursday in the election that is being held over six weeks and ends on Jan. 11.
The ruling army council, which took over from Mubarak, has said it will not let the transition be derailed and has pledged to hand power to an elected president by July. But protesters in the square want the army to return to barracks far sooner.
"God willing, we will complete the revolution by Jan. 25 by bringing down the army council," said 25-year-old protester Mahmoud, who declined to give his full name. The uprising against Mubarak began on Jan. 25 and lasted 18 days.
Near where he spoke, the authorities have erected walls of concrete blocks, barring access from the square on roads leading to the parliament, the cabinet and Interior Ministry where violence has been the most fierce.
A few hundred hardy protesters were still in and around the square on Wednesday, surrounded by streets strewn with rocks exchanged between them and security forces. Some protesters held up bullets and cartridge cases they say were used against them.
Traffic passed through other parts of the big square.
The clashes have driven a wedge between those determined to stay on the streets and other Egyptians frustrated by the turmoil, which has damaged the economy, and now desperate for a return of order. Many still see the army as the only institution capable of achieving this.
"All demonstrations should stop to end this violence until we finish elections and elect a president then all the demonstrators can voice their concerns through members of parliament," said Erian Saleeb, 64, who works in the tourist industry, which has been hammered as visitors have stayed away.
But many have been shocked by images of police and soldiers hitting protesters with batons even after they fell to the ground and, in one case, dragging a woman lying on the ground by her shirt that exposed her underwear, and then kicking her.
U.S. Secretary of State Hillary Clinton this week referred to that incident of the beaten woman as "particularly shocking" and cited other cases of women protesters sexually assaulted.
"This systematic degradation of Egyptian women dishonours the revolution, disgraces the state and its uniform and is not worthy of a great people," she said, in some of the strongest U.S. criticism of Egypt's new rulers.
The United States, which Egypt under Mubarak as a staunch ally, gives Cairo $1.3 billion a year in military aid.
About 2,000 women marched to Tahrir on Tuesday dressed in black and flanked by male protesters who vowed to protect them from harassment. "The women of Egypt are a red line!" they chanted.
"What happened to the girl who was stripped and dragged was sheer savagery. We cannot be silent about this. I want someone from the military council to admit responsibility," said Sarah Rifaat, a 27-year-old environmentalist.
In a statement, the army council apologised, saying it "respects and appreciates Egyptian women and their right to protest and fully participate in political life". An army general said it was an isolated case and under investigation.
But other generals and their advisers have condemned the pro-democracy demonstrators, who they accuse of wreaking havoc.
Retired general Abdel Moneim Kato, an army adviser, told the daily al-Shorouk that anyone involved in incidents such as the burning of a building near Tahrir that held historic archives was "a vagrant who should be burnt in Hitler's incinerators."
One opposition group that has lowered its profile in the protests is the once-banned Muslim Brotherhood, whose Freedom and Justice Party now leads the election results after the first round, followed by ultra-conservative Salafi Islamists.
A large percentage of the individual - rather than party list - seats up for grabs in the run-offs will be contested between Muslim Brotherhood and Salafi candidates. Egypt's system involves a mixture of party lists and individual candidates.
Analysts say the Brotherhood has kept a low profile as it is determined to see the vote completed, putting it in a commanding position in the new assembly and securing its place in mainstream politics for the first time in its 83-year history.
Washington has reached out to Islamists in a shift in approach since the summer. A senior U.S. diplomat met Islamist and other newly elected members of parliament in the northern city of Alexandria, the embassy said on Tuesday. (Writing by Edmund Blair; Editing by Mark Heinrich)