Bidirectional voltage and current switch using mosfets
The circuit works fine as a high side power switch with bidirectional isolation when off. For example, if you use it with a battery, it can prevent the battery from both charging and discharging.
But it cannot turn on when both A and B nodes are near or below ground. The reason is that the sources will be pulled down to ground and if A and B are both also grounded (or negative) then the diodes will not be forward biased and Vgs will be zero.
As is, this does not work as a switch for a signal unless you are able to pull the gates lower than the lowest voltage at the most negative swing of the signal.
I did once use a PMOS as an audio signal switch for the headphone jack in a toy. But in that case the signal was only a few hundred mV near midrail, and I used a very low Vgs(th) PMOS. It was tight but I got away with it.
The reason it is cutting off the lower half of your signal is that the voltage you are attempting to switch is too low to keep the FET's on. It is cutting off the part of your waveform that is below Vgs(th). Well, the cutoff may not be exactly at the threshold. It would be the threshold plus a little bit, let us say.