I've worked on a few fingering patterns for double stops in thirds, but always in the first inversion. Then I realized that I might be able to invert. Is there a reason this won't work?
On guitar, thirds are easy as they live on adjacent strings. Their inversions, known as sixths, are easier to play on strings that have another string in between - like 6/4, 5/3, 3/1.
They sound as good as thirds because essentially they are the same notes, just upside down.
The slight problem is how to play them. With thirds, one plectrum stroke across both does the job. A way around this is to mute the intervening string, and pluck all three. Same way as a lot of people play octaves on guitar.
The other is to finger pick each, either with finger/thumb, finger/finger, or pick/finger.
Yes, absolutely you can play strings of sixths melodically. I've seen it in transcripts from Charlie Christian. It's easy to get under your fingers because the pattern is the same across all the strings.
For each of the low 4 strings E A D G, the major sixth is two strings up and either on the same fret or one fret lower. And the minor sixth is one (more) fret lower.
The ring finger barre can be useful is making smooth runs with these.