The scope projects an image of a 10 volt signal because it is unaware that the point of reference has shifted. ( when it shows -5 on the display it actually measures 0 volts but with its 'ground' tied to +5 volts... so it shows '-5'. When it shows +5 volts its 'ground' is tied to 0 volts. )
When being teached this, i misunderstood that part because the gates only switch through their 5V and 0V rails (for TTL, or Logic High and Logic Low), making the whole thing Logic ground referenced, but the catch here is that the LCD/load is itself not ground referenced, it is connected across input and output, making the crystals following a +5 -> -5V span in potential. Of course the 10V are not present at the same time, as in a 10VDC source, they are only present when transitioning/comparing two halfes. Like in Simons example for mains AC. The 620VPeak-to-Peak are not present concurrently, but between half waves, making it a possible scenario to come across that voltage e.g. between two charged capacitors in one of
those voltage doubler configurations. Closest to the example at hand might be the Cross-coupled switched capacitors.
Depending on logic family you might get a different power output per edge as well, e.g. if you use a logic family with a high impedance high level the ability to drive a load changes.
I still wonder how much charge needs to be shifted in such a voltage doubler, and in case a part of the circuit becomes ground referenced. Probably only the parasitic capacity of this artificial/virtual ground between driver and load, which sees the same 10V potential? Now you attach an oscilloscope to see that waveform (assuming the circuit runs of an isolated supply)
a) direct, passive probe
b) with a differential probe
Does a) mean the gate needs to push the rest of the circuit ground to +5V and the included capacities to make that -5V reference shift, now that the ground reference shifts at the logic gate? Might be some fun to measure the current of the gate´s supply while it switches.