Well, that IS an unusual motor design. I cannot imagine there are many other examples of the same motor still surviving so its definitely worth restoring. Thanks for posting the pictures too. To answer the original question, I would say that you could not control its speed running it off a VFD. I might operate happily over a limited range of frequency (say 40 to 60Hz) if you had a suitable single phase supply.
I wonder what advantages the original manufacturer saw in that method of construction, it must have been more expensive to produce and I am at a loss to see any functional benefits, in fact I can only see disadvantages.
Anyone got any thoughts on the following points?
Purpose of the ring of small holes in the outermost contact ring (cooling maybe?)
If the device between the windings and the contacts rings is the centrifugal switch, its not going to help the rotor balance and its adds to the motor shaft overall length.
I presume there are two slip ring connections (carrying L and N) so the start winding is connected via the centrifugal switch.
Nowadays lots of motors have the windings on the inner part of the motor but the difference is that its the inner that is stationary (so called 'outrunner' motors). Papst, amongst others make mains voltage, single and multiphase motors with an outer rotor (once very popular as tape recorder capstan drives) but the gain in performance given by the flywheel effect of a heavy outer rotor is no benefit as the motors get larger or more powerful as its harder to keep the enclosed windings cool.
Ian P