Only 10 Iranian soldiers and 23 Iraqi soldiers were positively identified.
The ICRC, which supervised the exchange, said in August that thousands of Iraqi and Iranians are still missing, 20 years after the end of the war, in which one million people were killed on both sides.
It was the first handover since May 2003, when a US-led invasion had just toppled the government of Saddam Hussein.
Wooden coffins were brought from both sides before being draped with their respective flags and carried across by Iraqi and Iranian honour guards in crisp white uniforms.
Unresolved case
Mohammed Baghban, the Iranian consul in Basra, said: "We want to pursue this long unresolved humanitarian case until it is totally closed.
"There will be more remains to be handed over because there are still people missing."
On the Iranian side of the border, women in long black robes knelt and wept over the coffins of their loved ones.
On the Iraqi side, a military brass band played a procession as the long line of flag-draped coffins passed.
The war began when Iraq invaded Iran in September 1980 and ended in August 1988, after both sides accepted a ceasefire sponsored by the UN.
Grave unearthed
In another development, Iraqi officials announced on Sunday the discovery of a mass grave containing the bodies of 33 men, women and children in a former Sunni stronghold north of Baghdad.
The bodies were discovered near Khalis in Diyala, Amer Riffat, a member of the provincial council, said.
A police officer said the bodies showed signs of being blindfolded and shot.
Police said they believe the victims were killed by al-Qaeda in Iraq fighters who wielded sway in the area before a recent downturn in violence.