An explanation referrring back to ancient history, a bit simplified...
The ancient Lorenz approach system worked by transmitting two beams, one slightly to one side of the runway, modulated with dots, and one to the other side, modulated with dashes. If you were on the centre line, dots and dashes merged into a continuous tone.
The moment you can visualise this, you'll see that the attitude of the plane (even inverted) had nothing to do with it. If you were on the 'dots' side, you'd hear dots even with your head down.
The LOC and G/S system works somewhat differently... but the principle remains the same, it shows your position relative to the runway centre line and glidepath.
Now the kicker.... if you fly upside down, and hence you've turned the instrument upside down (think of the old Zero Reader), of course the indication is the wrong way round as welll
Think about it...
In normal flight, if the LOC needle is to the left, you turn to the left (towards the needle) to get back on the centreline.
In inverted flight, if the LOC needle is to the left, you now have to turn to the right to get back on the centreline.
Have fun.
CJ