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On page 12 of this manual:

https://www.scribd.com/document/176382158/Kitchen-Aid-K45-Service-Manual-For-Hobart-Made-Vintage-Mixers

The schematic shows the center tap of the resistor connected electrically to a "T" shaped plate. When the contacts of the plate close it furnishes a direct path of current from one contact to the other through the motor with a parallel path through the resistor (180 Ohms main terminals, 90 Ohms center tap to either main terminal). When the contacts open all current must pass through the resistor. Speed regulation is accomplished by the rapid opening and closing of the contacts which rapidly places and removes the 180 Ohm resistor in series with the motor.

I know the capacitor is for spark suppression.

When the contacts are open no current can flow from the resistor center tap to the "T" Plate and when the contacts are closed far more current flows directly through them that through any pathway of the resistor.

Why is this center tap connection there?

My guess is that this center tap to the main contact plate somehow reduces the tendency for sparking but I would like a definite answer.

BTW: Old Sunbeams used the same circuit arrangement except with the no center tapped resistor.

Thanks!!

Interesting question! Yes, switching between 90 and 180 ohms would be much quieter electrically than on/off action. It would also reduce the amount of sparking and prolong the life of the contacts and the motor. A definitive answer is unlikely to surface because the designers must be long since retired.

Thank You!

Yes, it may be almost lost knowledge. Perhaps I can find something in an older motor control book.

Another possible reason is the tightening up of EMC regulations, which started in the US around 1950 and were strengthened in 1970ish. An on/off motor speed regulator would have created a lot of conducted and radiated noise whereas the 90/180 type would have been better. ffi: https://www.electronics-notes.com/articles/analogue_circuits/emc-emi-electromagnetic-interference-compatibility/what-is-emc-basics-tutorial.php