For ease of further discussion, which must be very illuminating to the thread visitors, let us just look at the wing down case where we have carefully adjusted the bank to result in a centralised rudder whilst we maintain balanced straight flight.
Ah, perhaps this is the cause of the crossed wires.
If you hold the wings level and use rudder to correct the asymmetry, the aircraft will slip towards the
dead engine.
If you use wing down to correct the asymmetry to the extent that you're able to centralise the rudder, the aircraft will slip towards the
live engine.
Between the two is a zero-sideslip, wing-down, non-central-rudder scenario.