Why does Light Yagami shuffle the two death notes? Light Yagami gives a death note to Rem and asks Rem to give it to Ryuk. Then Ryuk drops the book according to Light's order. Light then takes back Misa's book while doing something similar with his own book.

Why did he do this?
If I recall correctly it was a part of Light's grand scheme to fully exonerate him as being Kira in the eyes of the task force and to kill L.
Only Light's Death Note had the rules written in English (see comments) and he added the fake rules in the back about needing to write a name very 13 days and if you destroy the Death note all those who touched it will die.

The Fake Death Note Rules are two rules that Light Yagami persuades the Shinigami Ryuk to write into the Death Note as part of a plan to clear him of suspicion. These rules ultimately lead to his downfall.

If the user of the Death Note fails to consecutively write names within 13 days of each other, then the user will die.

If a person makes this notebook unusable by tearing it up or burning it, all the humans who have touched the Death Note will die.



Source: Fake Death Note Rules
Ownership of this Death Note had to go to one of the Shinigami to then go to one of the Yotsuba Group (or any big company) and trusting L's detective work he would find the new Kira and recover the Death Note.
To do this Light had to have a second Death Note on him as he gives up ownership of his in order to retain his memories and claim ownership of the new Death Note when it's ownership is given up by one of the Shinigami. his new one he can give up ownership when he wanted to &quot;get rid of this pride&quot; while in imprisoned (ie. get rid of ownership and erasing his memories).
Now it was important for Rem to be the Shinigami of this Death Note that L recovered because if Rem didn't play along Misa's life would be in danger as such the fake rules wouldn't be exposed. Of course Light knew L well enough that L would still pursue Misa and seeking her execution by proving she was the Second Kira, forcing Rem's hand to then kill L. L would then be killed by the Shinigami everyone knew about and not by Kira and it would also kill Rem having intervened in Misa's death.