Sending signals both ways on one coax is very possible, but like Wombat says, it really does need a splitter/combiner at each end. For this to work reliably, the power offered at both ends needs to be comparable, which implies a receiver/booster in the radio control bit. There would, for simplicity's sake (!), need to be both a separate aerial for send and receive. If the two signals were widely separated in frequency, the input circuits MIGHT separate out the unwanted signal generated by the output being fed into them. Both might would need some impedance matching to work with the chosen cable.
All this is just musings thinking back to the odd bits of line transmission theory I had to do at tech about 45 years ago. Back then GigaHertz hadn't even made it into Science Fiction, let alone the classroom.