why would one use a current transformer with both live and neutral lines?
Based on the wires shown, this is part of a ground fault detection circuit or GFCI transformer. The red and blue wires only output the difference between 'Hot' and neutral current as an AC voltage. The idea is that if the hot wire is leaking current to ground from bad insulation or if someone touches the hot wire or a shorted appliance, enough imbalance will occur to trip the circuit OFF. With balanced loads that return all source current back through neutral, the currents cancel each other out, so the red-blue pair have essentially zero volts AC on them. Usually the trip point is 1mA of imbalance to trip these GFCI circuits. With a fast response time, it may also trip on 'arc' faults where a leakage in the insulation of downstream wiring is wet, or the wire is pinched by a staple or nail, etc, and starts to arc with small current spikes before the insulation burns enough to cause a fire. Outside outlets that are not rain-proofed can be prone to arc faults. Sauna steam rooms do not have AC power outlets as the thick steam would create an arc fault on the first day of operation.