From the wikipedia article on sine waves:
The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and magnitude. It is the only periodic waveform that has this property. This property leads to its importance in Fourier analysis and makes it acoustically unique.
I don't follow these statements. If you add two sine waves together with identical phase, isn't the amplitude doubled? Sure, it's still a sine wave, but isn't the same true of a square wave? If I add two square waves with equal phase, it's still a square wave, just with doubled amplitude. It has retained its shape. What am I missing?