How and why was the 501st Legion on Mandalore and Coruscant at the same time?

Question

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 Wookieepedia (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,[2] Rex removed his inhibitor chip, an inorganic brain implant that compelled each clone trooper to comply with Order 66.[17] Both Rex and Tano escaped from Mandalore, as did Maul, while a fresh wave of violence gripped the Mandalorian homeworld.[2]

[2] is Ahsoka

[17] 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?

Answer 1

We don't have canon info about the size of a legion, but in Legends it is roughly around 10 000 men, similar to a division in large number of real armies here on Earth, especially to World War 2 divisions, and we know Lucas did get a lot of inspiration from WW2. Vader had approximately one battalion with him in that attack on the Jedi Temple, which suggests that 501st Legion was split up, again something that does happen in real life, assuming that one battalion in SW universe has similar size to real life battalions (around 1000 men).

Moving the whole elite 501st Legion to Coruscant without apparent reason would probably raise suspicion. But moving one or two battalions in disguise of routine rotation, rest and replenishment would go undetected. Palpatine probably pre-calculated how many troops would be necessary for planned attack, so he acted accordingly.

Answer 2

Now that the new clone wars episodes have come out, we know that

Anakin, Obi-Wan and Ahsoka were preparing to invade Mandalore to take it from Maul when Anakin and Obi-Wan were called away to rescue the Chancellor at the Battle of Coruscant. They Jedi planned to just abandon Mandalore until Ahsoka called them on their politicking, at which point they decided to split the 501st, and send part of it to Coruscant and leave the remainder to invade Mandalore lead by the newly promoted Commander Rex.