The Christian Liberation Movement called on the Cuban military junta Monday to carry out a “transparent’’ investigation of the deaths of its founder Oswaldo Payá Sardiñas, a champion of civil society, and dissident Harold Cepero Escalante, who died in a controversial car crash.

It’s still not clear what happened Sunday afternoon on a pot-holed road in eastern Cuba as Payá, 60, who fought for the rights of the Cuban people for more than two decades, and Cepero, 31, the Movement’s youth leader, traveled in a rental car with two European companions.

Payá, who lived in Havana, was best known for his role in organizing the Varela Project, a signature-gathering drive in support of a referendum on laws to guarantee freedom of speech and other civil rights.

The two international supporters were identified as Ángel Carromero Barrios and Jens Aron Modig. Both sustained minor injuries. Spanish media reported that Carromero is a leader of the Spanish Popular Party’s youth organization, Nuevas Generaciónes, and that Modig is president of the Swedish Christian Democrat Youth League, wing of Sweden’s ruling alliance.

Spanish news agency EFE, citing sources, said Carromero was driving.

The Cuban government said the driver of the rental car lost control and hit a tree at 1:50 p.m. local time Sunday in La Gavina, a town about 14 miles outside Bayamo, the capital city of Granma province.

But Rosa María Payá, the dissident leader’s daughter, said their car was struck by another vehicle. In a recording on Payá’s official website, she said, “The information we received from the boys in the car with him is that a car was trying to push them off the road, ramming them at every moment. So we think — we are convinced — that they wanted to harm them and ended up killing my father.”

In a statement Monday, the Christian Liberation Movement said “the circumstances of these deaths have not been cleared up and are open to hypothesis’’ and it demanded a “transparent’’ investigation.

Some members of the Movement remained suspicious. Julio Hernandez, the group’s Miami representative, said Payá told him three weeks ago that his car had been wrecked and flipped when another vehicle hit him in Havana. “He said he was alive by a miracle,” Hernandez told El Nuevo Herald.

However, Yoandris Montoya and Felix Rivero, two dissidents from Bayamo who were sent to the crash site by the Cuban Commission for Human Rights and National Reconciliation, said two people who claimed to have witnessed the crash told them the vehicle ran off the road on its own to avoid a pothole, tumbled and hit a tree. The stretch of road was under repair.

In an unusual step, Granma, the official newspaper of the Communist Party of Cuba, reported the deaths Monday on page six and said the crash was under investigation.

The story called the incident “a regrettable traffic accident.” Although it mentioned the victims by name, it described them simply as “Cuban citizens.’’

Ofelia Acevedo, Payá’s widow, said the Christian Liberation Movement “will continue its peaceful fight until all Cubans win the rights we have by law. My husband dedicated his life to this ideal until the end.

“From eternity,’’ she said, he will “encourage and accompany us until truth and justice make our dear island an authentic home for all Cubans.’’