Why is it that when I play a major chord on my guitar, the tuner shows the correct chord, even if the root isn't in the bass ?
Depending on how your tuner detects frequency, it's possible that this is because of the resultant tone phenomenon.
If we play two sinewaves, one at 100 Hz (about G2), and another a musical fifth above at 150Hz, and then add the waveforms together, the summed result is a waveform whose actual frequency is the difference between those frequencies - ie. 50 Hz (about G1).
If we play a note at 100 Hz and another note 25Hz above, which is a Major third above, the resultant tone is at 25Hz - around G0.
This will even work with some inversions. If we play 150Hz and a G an octave higher - a (slightly flat) G3 at 200 Hz, the resultant tone is still 50Hz and still a G.
I don't think this would explain it if your tuner can ALWAYS pick out the right chord - especially as real notes aren't sinewaves, so it won't work as simply as I outlined - but it may explain why some voicings 'work'.