When the Earth's magnetic field reverses, does its rotation also switch?

No it doesn't, because although the magnetic reversals can be easily identified in the rock, there is no evidence whatever that the rotation of the Earth falls to zero, then spins back up. This is impossible from simple physics, without turning the Earth's surface molten because of the enormous heat dissipation that would result. The figure above shows how the magnetic field has been changing over the last 13,000 years and how we are currently in a state of declining field strength which may reach a minimum in another 5,000 years.

If you want to learn more about what happens during a magnetic reversal, read the IMAGE satelite article Magnetic Reversals: Back to the future? which discusses all of the details and likely impacts.