* Mursi says "Down with military rule"
* Mursi charged with inciting violence
* Rights groups say trial seen as test for authorities
* Government warns Brotherhood over protests
By Yasmine Saleh and Yara Bayoumy
CAIRO, Nov 4 (Reuters) - Ousted Egyptian leader Mohamed Mursi, given his first public forum since his overthrow in a trial where he could face execution, declared on Monday he was still Egypt's legitimate president and shouted: "Down with military rule!"
Mursi, an Islamist who was toppled by the army in July after mass protests against him, spoke with anger and passion, interrupting the first day of his trial repeatedly from his cage during an unruly hearing that the judge adjourned to Jan. 8.
State television aired brief footage of Mursi, the first public sighting of the president since his overthrow in July. Mursi, Egypt's first freely elected president, had been kept in an undisclosed location since then.
"I am Dr. Mohamed Mursi. I am president of the republic," said Mursi.
Inside the courtroom, anti-Mursi Egyptian journalists chanted "execution", "execution" as the deposed leader did his best to challenge the authority of the court, shouting repeatedly at the judge whose legitimacy he refused to accept.
"We are in a state, not a (military) camp. Down down with military rule," said Mursi. "I am a witness that what is happening is a part of a military coup. I ask the Egyptian judiciary to not act as a cover for the military coup."
The judge repeatedly asked Mursi to stop giving long speeches. "Please answer the question, do you agree to have a lawyer representing you?" judge Ahmed Sabry said.
Opponents of Egypt's army-backed government deride what they call a "show trial" as part of a campaign to crush Mursi's Muslim Brotherhood movement and revive the police state of Hosni Mubarak's 30-year rule that ended in a 2011 popular revolt.
Hundreds of people were killed in the months that followed Mursi's overthrow, including many hundreds shot dead by police and troops who cleared out a weeks-long protest vigil by Mursi's supporters. Thousands of followers have been rounded up.
Egypt has become fiercely divided, with state media lionising the military and police for their crackdown on "terrorists", while the Brotherhood, once the country's most powerful political force, has retreated to the shadows where it spent more than 80 years as an underground movement.
Mursi, 62, who like many Islamists was also jailed under Mubarak, now faces charges of inciting violence that could carry the death penalty.
It is the second time Egypt has put an ousted president on trial since 2011, and taking place in the same venue - a police academy hall - where Mubarak has faced retrial over his conviction for complicity in killing protesters.
Mursi and 14 other Islamists face charges of inciting violence relating to the deaths of about a dozen people in clashes outside the presidential palace in December after Mursi enraged his opponents with a decree expanding his powers.
After stepping out of a white van and buttoning his jacket, he appeared in a cage in the courtroom beside other Islamist defendants, who were in white prison garb. They applauded when Mursi arrived, gave the Brotherhood's four-fingered salute, and at times turned their backs on the court.
"This trial is illegitimate," said Mursi, who was dressed in a dark suit. "This is a criminal military coup."
Hundreds of Mursi's supporters gathered outside the court building. One sign read: "The people's will has been raped".
Trial proceedings were not aired on state television and journalists were barred from bringing telephones into the courtroom. Senior Brotherhood figures among the defendants used the chance to tell reporters they had been mistreated.
"I have been kept in my cell for 60 days," Brotherhood leader Mohamed El-Beltagi told Reuters in the courtroom from inside a cage holding defendants. "I have been held under water in my cell and this has happened to other members."
Another Islamist in the cage, Alaa Hamza, said he was tortured and lifted his shirt to show reporters what he said were torture marks.
After the hearing, Mursi was taken to Borg al-Arab prison in Alexandria.
The military establishment's return to the forefront of power prompted Washington to cut some military aid, although Washington has not said whether the overthrow was a "coup", language that would require it to halt aid to one of its biggest clients. U.S. Secretary of State John Kerry, visiting Cairo on Sunday, expressed guarded optimism about a return to democracy.
The uprising that toppled Mubarak in 2011 had raised hopes that Egypt would embrace democracy and human rights and eventually enjoy economic prosperity.
Instead, the power struggle between the Brotherhood and the army-backed establishment has created more uncertainty in the country of 85 million which has a peace treaty with Israel and controls the Suez Canal. Tourism and investment have collapsed.
The Brotherhood won repeated elections since Mubarak's fall. But millions of Egyptians grew disillusioned with Mursi's troubled one-year rule and took to the streets to demand his resignation. They accused Mursi of usurping power and mismanaging the economy, allegations he denied.
"We didn't see as much misery in the 30 years of Mubarak as much as we saw in one year of Mursi," said Ali, a driver who was sipping morning tea at a cafe in downtown Cairo.
"He fooled us with his year in power."
The army, saying it was responding to the will of the people, deposed Mursi and announced a political road map it said would lead to free and fair elections.
But the promises have not reassured Western allies, who had hoped six decades of rule by military men would be broken. Army chief Abdel Fattah al-Sisi, who toppled Mursi, is very popular, and few doubt he would win if he runs for president.
The Brotherhood maintains Mursi's removal was a coup that reversed the democratic gains made after Mubarak's overthrow.
Mohamed Damaty, a volunteer defence lawyer for Mursi, said:
"It is clear that the goal of this trial as well as any action against the Muslim Brotherhood is to wipe out the group as well as any Islamist movements from political life."