The square wave has only odd harmonics. The interesting similarity to the sawtooth wave is that each harmonic decreases in the same manner except the third harmonic is ½ the amplitude of the fundamental, with the fifth harmonic at 1/3 the amplitude of the fundamental, and continuing along in that manner. This harmonic structure gives the square wave a little more bite to the sound, which can be heard here.
100hz Square Wave
500hz Square Wave
1000hz Square Wave
Other Soundwave Examples:
These main four waves can be seen in most synthesizers, DAWs, and testing equipment. There are some variations that include the sawtooth wave going in opposite directions and the lopsided square wave (being larger on the top and smaller on the bottom or vice versa). Depending on the device these waves can be added together to create complex waveforms, can be used to modulate a signal, or can create the pattern for a pulsating effect.