Why do differential signals need an inverse wave?
why does a differential signal need an inverse input

They don't. What makes &quot;differential&quot; signals work is that the impedance of both wires is equal (balanced) which is necessary for noise to be picked up as common mode which can then be rejected. If the impedance of both wires is different, then some of the noise will be picked up as differential signal.
For example if you use a current sense resistor you'll have better noise rejection by routing the sense lines as a differential pair. In this case there is no inverted signal, but routing as a differential (and balanced) pair ensures the noise picked up by both lines is common mode and rejected by the diff amp.

Using a positive and inverted signal just doubles the available signal amplitude, which gives you a bit of extra signal to noise ratio or headroom. It also minimizes some distortion mechanisms in the receiver (ie, odd order distortion).