* Swearing-in delayed as Chavez remains in Cuban hospital
* Supporters gather in streets around presidential palace
* Three left-wing Latin American leaders attend events
By Daniel Wallis and Brian Ellsworth
CARACAS, Jan 10 (Reuters) - Venezuelan President Hugo Chavez remained on his sickbed in Cuba on Thursday while thousands of supporters rallied in his honor on the day he should have been sworn in for a new six-year term in the South American OPEC nation.
The postponement of the inauguration, a first in Venezuelan history, has laid bare the gravity of Chavez's condition after complications from a fourth cancer operation in his pelvic area.
It has also left his chosen heir, Vice President Nicolas Maduro - a former bus driver who shares his boss's radical socialist views - in charge of day-to-day government until there is clarity over whether Chavez will recover.
The president, whose legendary energy and garrulous dominance of the airwaves had often made him seem omnipresent in Venezuela since taking power in 1999, has not been seen in public nor heard from since his surgery on Dec. 11.
"Only God knows what will happen," William Medina, a 49-year-old worker, told Reuters amid crowds of red-clad supporters milling around the presidential palace, many waving banners and posters bearing their hero's face.
"But we are ready to take on what he taught us, because each one of us is a Chavez. We are ready to continue with socialism, because that is the only way to save planet Earth."
Venezuela's 29 million people are anxiously watching what could be the last chapter in the extraordinary life of Chavez, who grew up in a rural shack and went on to become one of the world's best-known and most controversial heads of state.
The saga also has huge implications for the likes of Cuba and other leftist allies in Latin America that have benefited for years from Chavez's subsidized oil and other largesse.
A clutch of foreign friends, including the presidents of Uruguay, Bolivia, and Nicaragua, attended Thursday's events in Caracas despite Chavez's absence.
" There's a man who is battling for his life; he is in your hearts and that's what matters," said Uruguayan President Jose Mujica from a stage outside the Miraflores palace, the scene of dramatic moments in Chavez's rule including his return from a failed 2002 coup and euphoric speeches after election victories.
Sukhoi jets, which Venezuela bought from Russia after a diplomatic dispute with Washington, rumbled above the demonstration, drawing a roar of approval from the crowd.
Venezuela's opposition leaders are furious at what they see as a Cuban-inspired manipulation of the constitution by Maduro, Diosdado Cabello, who is the head of the National Assembly, and other Chavez allies aimed at preventing the naming of a caretaker president due to Chavez's absence on Thursday.
Henrique Capriles, who lost October's presidential election to Chavez, said the opposition had no plans to risk violence by encouraging supporters to hold a counter-demonstration.
"Who wins from a conflict scenario?" he asked. "They win, the pseudo-leaders who are not the owners of the country."
A top Venezuelan military officer told state TV the borders were being reinforced and security forces were patrolling to bring people "a sense of peace and tranquility."
Little is known about Chavez's actual condition, with terse official updates confirming he has a severe lung infection but offering few details. Speculation is rife on Twitter that he may be on life support or at risk of major organ failure.
He has undergone four operations, as well as weeks of chemotherapy and radiation treatment, since being diagnosed with an undisclosed type of cancer in his pelvic area in June 2011.
He looked to have staged a remarkable recovery last year, winning a new six-year term in October. But within weeks of his victory he returned to Havana for more treatment.
'REVOLUTION MUST CONTINUE'
In contrast with previous trips to Cuba, the government has not released any photos or video of him recovering, and Chavez has not made any phone calls home to state media, fueling the impression that his condition is dire.
Though supporters maintain vigils and express hope he will recover, there appears to be a growing acceptance he may not, and a slow adjustment to the idea of a post-Chavez Venezuela.
"We are all necessary but nobody should be irreplaceable and the revolutionary process in our America must continue," said his friend and close ally, Ecuador's President Rafael Correa.
Though often viewed in the West as a clownish autocrat, Chavez has a kinder image in developing nations where many admire his defiance of the United States and efforts to improve the lives of Venezuela's poor.
At home, Chavez has a cult-like appeal for many in the slums due to his "anti-imperialist" rhetoric, his pumping of crude oil revenue into welfare projects, and his own humble background.
But Venezuelan opponents say he has squandered an unprecedented bonanza of oil money with misguided policies while taking control of state institutions.
Should Chavez die or step down, a new election would be called and it would likely pit Maduro against opposition leader Capriles, the 40-year-old governor of Miranda state.
Past polls have shown Capriles to be more popular than the vice president, but Maduro would likely be hard to beat given Chavez's personal blessing and the emotional outpouring from supporters if the president were forced to leave office.
In a sign a Maduro-led government would continue Chavez's tough treatment of foes, a state regulator has started punitive proceedings against opposition TV station Globovision for causing "anxiety" with its coverage of the president's health.
Investors hope for a more business-friendly government in Venezuela, and this has caused prices of its widely traded bonds to soar over the last few weeks amid Chavez's health concerns. But they dipped this week as expectations of a quick change apparently dimmed.