Work began after a post-mortem examination was carried out in a bid to establish what killed the gentle giant.
The 44-foot long male sperm whale was found washed up on the beach close to Green Lane by an early morning walker yesterday.
Extensive efforts were made to save the massive mammal but it later died.
The spectacle was played out in front of hundreds of spectators who had come to see the stricken animal.
Now work is under way to dispose of the 35-tonne creature which, if left, could become a health hazard.
The area has been cordoned off and a screen erected while the carcass is taken apart by specialist contractors.
Steve Goldswain, Redcar and Cleveland Council’s cabinet member for community safety, said: “This is a complex situation which must be handled carefully and sensitively, and there is a lot of expertise among the teams carrying out this operation.”
There has been much speculation about the cause of the whale’s death.
It has even been suggested that it might be linked to a recent controlled explosion after a German Second World War mine was found off the coast.
Kevin Kneebone, managing director of explosive specialist Bactec, confirmed the firm carried out a detonation about 10km offshore from Redcar on May 21.
He said that such explosions can affect marine life, “which is why we put the correct environmental procedures in place”.
But he ruled out any link, saying they were “very experienced” at such operations and had never had “any link to any adverse effect on marine life”.