about your specific case: the twisted pair should work well, however since you are in a cave you could also use sonic communication quite easily, like how submarines do it.
however as a answer on the general question
Yes you can communicate with one wire. you don't even need a "ground"
note this method relies on a parasitic grounding effect which typically happens on places like earth.
you can use the signal and a capacitor,
instead of measuring the current state you measure the changes, so you measure it kind of like you do with a capacitor isolated circuit like in medical or audio equipment.
so when it is High first on when the capacitor charges it will send a positive pressure in the circuit, then it might go back to around 0, but that isn't bad, since even if it goes back to 0 there will be a negative pressure spike generated when the voltage of the signal is brought to low. so you get a pulse when the signal turns high and when it turns low
this will work well and you only need one wire and a capacitor to decouple it so you get the relative voltage change rather than the exact voltage.
however it doesn't end here because you can also use diodes to rectify the signal, you might not even need a capacitor that way, however you can use one to get a refference high and low voltage, when doing it like that you can even use most common communication protocols which need a ground.
protocol the data would based on the frequency, so you can either use the pulse length to determine if it is a 1 or 0 or if you monitor something simple like sound or a analog value then you can directly use that as the PWM, this is actually quite simple to how microphone amplifiers work, however now you have much more voltage and current and don't need a ground.
in short the most easy way to use a electric single wire for communication is by using the voltage change instead of the voltage itself by decoupling it with a transistor. if desired a reference voltage can be made using 2 diodes and a capacitor, since the diodes have a small voltage drop anything above the reference voltage high is high, and anything below the voltage drop low is low. for example while using 12V for communication the reference voltage for low would be around 0.7V and for high around 11.3V. this actually makes it more easy to work with. however all of that is rather unnecessary, since you don't need the reference voltage in most cases when you decouple it and measure the change instead of the actual value, in some minimalist circuits it might make it more easy however.
you should use higher voltage for this to work properly, it will kind of automatically use the air and the earth as a ground, so it wouldn't work as well in space, you can see it kind of like gravity lines or magnetic fields while in theory they shouldn't as easily interfere due to many factors they still will do so. with a magnet as a example when you rotate a flying magnet it should go back and the other magnet shouldn't notice this, however due to the magnets mass and air resistance or other resistance, so the arm holding the measuring magnet will measure a magnetic peak force.
using a ground wire will make it a lot more stable and efficient however, it will give a much stronger signal.
when sending signals through long wires you also need higher voltages typically, and due to internal resistance and capacitance it will become a lot harder for the described method to work properly, when you connect both grounds to the earth however instead of relying on the parasitic ground effect it will be a lot more effective.