6ths are upside down 3rds.
And 3rds are upside down 6ths. They are two different forms of the same thing. 6ths complete all major and minor chords that are not in root position, and they are the outside of inverted augmented chords and diminished seven chords. They also are the outside of something called an augmented seven chord.
There is a PDF of 6ths coming soon.