He said that he and Aileen later met in person and he was told the truth then, and once he found out "everything fell into place".
He said: "My mother was constantly voyaging out to Cardonald [where his grandparents lived]. I once thought she wanted him to go down on his bended knee and beg forgiveness. No. She was in love with her own father, utterly and totally.
"I think he gave me that wonderful mixture of genes with his own daughter, the eldest of the family. That's what they do in Ireland. I presume it's what they do in unlettered places and lettered places. It's traditional, and nobody speaks about it."
His mother had tragically died while in hospital in the 1980s. She suffered from a mental illness which he believes was a result of the relationship.
Byrne split up with Swinton in 2005. The pair have two children together.
The artist has been with theatrical lighting designer Jeanine Davies since 2006.
He is known for designing the front cover of The Beatles album, The Beatles Ballads, and wrote the 1987 series Tutti Frutti, starring Robbie Coltrane and Emma Thompson.
Several of his paintings hang in the Scottish National Portrait Gallery.