I am not an expert in electronics. Hence, I am looking for a broad conceptual answer.
Is it possible to a directional Bandpass/Bandstop filter i.e. Bandpass in one port and Bandstop in another.
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2-wire telephony circuits handle signals travelling simultaneously in opposite directions and "figure out" which is the one they are interested in processing. They achieve this by using Wheatstone bridge circuits where the balancing impedance is the known and controlled impedance of the telephone wire. This is why when blowing into a handset microphone you don't get deafened by the noise coming from the earpiece. Tried and tested technology.
An extension of this principle is splitting the simultaneously occurring send and receive signals into forward and return channels. The circuit that does this is called a "hybrid" and they are used in 2 to 4 wire converters: -
The above circuit (from RF World) uses two hybrids to split forward and receive signals into simpler unidirectional channels so that conventional amplifiers can be applied and this, of course, makes a 2-wire, bidirectional line amplifier.
Is there a configuration where a circuit acts as Bandpass filter in one direction and Bandstop in other?
Using the idea above, put filter A in the forward channel and filter B in the return channel.