Conservative MPs have flexed their muscles on the first day of the new Parliament, with Liberal senator Cory Bernardi signing up every government backbench senator bar one to a motion that would re-write section 18C of the Racial Discrimination Act.

But the libertarian South Australian senator denied the move represented a direct challenge to Prime Minister Malcolm Turnbull's authority, even as he kick-started the push to re-write the law to drop "offend" or "insult" from the Act, while leaving in "intimidate" or "humiliate".

At the same time, Opposition Leader Bill Shorten put Mr Turnbull on the spot as he moved a motion that would have the Parliament restate its commitment to all Australians enjoying equal rights and deserving equal respect, "regardless of race, colour, creed or origin".

Mr Shorten's motion is identical to that proposed by former Labor leader Kim Beazley in 1996, soon after Pauline Hanson was elected for the first time, which was eventually put up by the former prime minister John Howard and backed by the then opposition leader. Mr Turnbull is said to be considering following Mr Howard's lead and putting forward the motion.