I understand they have an extra winding.
But I believe that extra winding needs a capacitor in series to provide the required phase-shift.

No. I have a number of them without a capacitor. A capacitor start motor does not need the centrifugal switch as far as I understand, but those with the switch do not need the capacitor.
The start winding is wound offset from the run winding to produce the phase shift to get it going.

 

The start winding is done, usually, with a different gage of wire and a different number of turns so that the different inductance produces a phase shift. Thew fact is that it was done this way for many many years, and we had to put up with less starting torque because the phase shift was not as great as with a capacitor. I have a few of those older motors still.
So keep in mind that just because you do not understand it does not mean it is not true.

 

those with the switch do not need the capacitor.

Some use a switch for the capacitor and winding.
The capacitor is designed only for momentary use, not continuous.

 

So keep in mind that just because you do not understand it does not mean it is not true.

Yes, master.

 

I understand they have an extra winding.
But I believe that extra winding needs a capacitor in series to provide the required phase-shift.

Not always, for smaller, low power motors, the second winding provides a certain shift in that part of the core to get it started, once running the natural lag through the stator lamination is sufficient.
In a similar fashion to where a start capacitor is either open or the centrifugal switch is.
And giving the shaft a spin at power on will run the motor, possible to spin in either direction,