Maybe I can explain it better.
At no signal, the voltage at the emitter due to DC current through the emitter resistor charges the capacitor to the same voltage (to ground) on the amp side, but no DC current can flow through the capacitor, so the speaker side of the capacitor is grounded through the speaker. No current flows throught the speaker and the coil is neutral.
Along comes a signal that causes the emitter voltage to swing more positive. Electrons rush out of the amp side of the capacitor and this attracts electrons to the speaker side of the cap. Electrons flow through the speaker from ground to the cap. The cone moves (pick a direction) say, inward.
Along comes the opposite polarity part of the signal and makes the emitter go less positive to ground.
Electrons rush back back into the amp side of the capacitor and that repels electrons out of the speaker side of the capacitor. They flow the opposite way through the speaker coil and the cone moves the other way.
When the audio signal stops and the amp goes quiescent again, the capacitor charges to whatever the emitter to ground voltage is, and stops passing current. The speaker doesn't "know" what the voltage on the other side of that capacitor is...the speake AC current stops flowing and the cone returns to neutral position.
cheers
Rob