Push-pull output stage and driving circuit function
This requires a crystal ball to imagine why the circuit was designed this way without you telling us where you found it. I don't think it is a great design but it could be useful as a floating high side Pch level shifting pulse driver if the 6 Ohm load was AC coupled to a high voltage gate for a negative spike to trigger an IGBT or FET 

I understand what waveforms to expect as the input square wave switches off each Vt=2~4V threshold FET with a DC restorer diode to each rail. Each gate turns on only for a pulse of a few μs due to T=C5*R5 = 470 nF*6R = 2.8 μs.

M2 is current-limited only by R5 in ground switching yet M1 is current-limited by R2 = 50 Ω  shunted the 6 Ω AC coupled load so the positive spike will be useless for triggering FETs at only 6/56*6 ~= +0.64V while the negative spike can be -6V.

So because of R2, I don't see this as being a useful bidirectional pulse output with a lot of redundant parts for just a negative AC coupled pulse.