Norbert Hofer of the Austrian Freedom Party (FPOe) and left-leaning independent Alexander Van der Bellen were debating ahead of Sunday's vote.
Mr Hofer hopes to become Europe's first far-right president since World War Two.
Opinion polls point to a tight race after a marathon campaign.
Mr Hofer successfully argued for a re-run of the vote, after a dispute over postal ballots when Mr Van der Bellen narrowly won in May.
Although the role is largely ceremonial, the Austrian president does have the power to dissolve the lower house of parliament, the National Council, and trigger elections.
Mr Van der Bellen said he was hurt that one of his opponent's aides had called his father a "Nazi". "You just don't do that," he said, whereupon he held up a photo of his father to the TV camera.
Mr Hofer called that move "a dirty foul". He said his father had also been insulted as a "Nazi" on social media. "Stop trying to attract sympathy in that way - the same thing has been done to me," he told Mr Van der Bellen.
Mr Hofer also alleged that Mr Van der Bellen had been a communist spy because of a report on weapons that he had written.
"You're lying again. This is ludicrous," Mr Van der Bellen said. He said spying was a capital offence and "such things are totally alien to me".
On security, Mr Hofer said any jihadist returning to Austria from a Middle East conflict zone should be stripped of his Austrian citizenship. Mr Van der Bellen disagreed with the proposal.
Mr Van der Bellen accused his rival of agitating for Austria to leave the EU - something that Mr Hofer denied.
Then Mr Van der Bellen said Mr Hofer had friendly ties to French nationalist leader Marine Le Pen - and Mr Hofer hit back by calling Mr Van der Bellen a friend of the late Cuban revolutionary Fidel Castro.
A new - and very long - German word was coined during the campaign. "Bundespraesidentenstichwahlwiederholungsverschiebung" means "the delay in the repetition of the presidential run-off vote".
Despite its length, the word is also a Twitter hashtag.
Norbert Hofer
Alexander Van der Bellen
Austria's presidential candidates went head-to-head on Thursday in a bitter TV debate in which both complained that their fathers had been labelled Nazis.