Bingowings said:

It was suggested earlier that Grievous takes his Jedi trophies and modifies them to look like Sith sabers, as a form of defilement ( in real world terms it would be like painting Nazi symbols on the graves of their victims).

Grievous takes Obi-Wan and Anakin's sabers at the beginning of the film.

Rather than making the added effort of taking their own weapons back they grab the nearest.

So Anakin ends up with a red bladed saber.

When Kenobi fights Grievous he retrieves Anakin's blue one.

That way Vader has a red blade (taken from Grievous) and Obi-Wan has Anakin's original blue saber (taken from Grievous) and the scene you describe is no longer necessary.

Alternatively Palpatine's saber doesn't go out the window and Anakin in his rush to save Palpatine leaves his saber behind (he spent much of the last film doing this sort of thing).

So he grabs Palpatine's red blade to disarm Windu.

Palpatine could then give him a new blade to go into the Temple.

But then presumably you have to change the hilt in Anakin's hand in every scene that he fights Obi-Wan.
It just seems like such a complicated route to go for such a cosmetic idea.