If the MACH III head is secured to the base from the back it will fit onto the Firefly and can be screwed in for carrying by the handle. This is what makes the MACH III and Firefly a MODULAR series - you can switch the heads between the bases. The Firefly is essentially an Electric MACH III.

If the MACH III head is secured to the base from the side it will fit onto the Firefly but it cannot be secured to the Firefly. This does not effect performance, but it does mean that you will not be able to carry the MACH by the handle.

The newer heads (new: secured to the base from the back) will fit the older MACH III but cannot be secured (by the screw) into the base. This does not effect the ability to spin, but it does mean that you will not be able to carry the MACH by the handle.