Opposition Leader Bill Shorten says he will continue to cooperate with the Royal Commission into union corruption.

But he has repeated demands for the release of all correspondence between commissioner Dyson Heydon and organisers of a Liberal Party fundraiser.

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Mr Heydon has withdrawn from the August 26 speech, telling organisers that if there was any possibility it could be described as a Liberal Party event he couldn’t do it while presiding over the royal commission.

“I think that for many Australians the fact that Mr Heydon was billed as a guest speaker at a Liberal Party fundraiser is the smoking gun of political bias,” Mr Shorten said on Friday.

Mr Heydon opened Friday’s hearing without mentioning the furore.

On Thursday he withdrew from the Sir Garfield Barwick address, telling organisers he couldn’t do it while presiding over the royal commission into trade union corruption.

Meanwhile, the ACTU has written to Prime Minister Tony Abbott asking him to end the unions royal commission.

In the letter, obtained by AAP, ACTU secretary Dave Oliver and president Ged Kearney say the commission has “no claim to independence, impartiality or adherence to due process and must be terminated immediately”.

They say Mr Heydon’s initial acceptance of the invitation had made his position “untenable”.

– with AAP