Is there an explanation as to why the 501st Legion is on Mandalore for the Siege of Mandalore AND on Coruscant during the initial stages of Order 66?
We See Darth Vader and the 501st Legion purge the Jedi Temple on Coruscant. (Episode III: Revenge of the Sith).
We do not know if the 501st is on Mandalore. We know that Captain Rex and Ahsoka are both on Mandalore during the Siege of Mandalore. (Ahsoka(Novel), Star Wars Rebels S2:E3)
From the Ahsoka Novel, we also know that Ahsoka was on Mandalore when Order 66 was executed.
From Wookiepedia (not sure how accurate that is) we also have this section (emphasis mine):
Nevertheless, the Republic's success did not last long. Chancellor Palpatine, who was in fact Darth Sidious, issued Order 66, which forced the Grand Army's clone troopers to execute their Jedi commanding officers. During the ensuing chaos on Mandalore, Tano was presented with the opportunity to kill Maul, but she instead chose to save Rex. Rather than execute Tano, Rex removed his inhibitor chip, an inorganic brain implant that compelled each clone trooper to comply with Order 66. Both Rex and Tano escaped from Mandalore, as did Maul, while a fresh wave of violence gripped the Mandalorian homeworld.
 is Ahsoka
 is Rebels Episode (S2:E3)
From the quote, it is clear that Order 66 was issued during the siege and that Rex was on Mandalore during the siege and the order being issued.
Is it possible that Rex was on a special assignment and the 501st was on Coruscant during the siege?