Interval Inversions
Triad inversions showed three notes that stay the same chord no matter which one is on the bottom. A bare interval - just two notes - has an inversion too, but it's really about distance rather than about the notes themselves: any interval's inversion is just whatever's left over after subtracting it from a full octave. Count a small distance up from a note and you get one interval; count the complementary large distance up from that same note and you get another - a different name entirely, and yet the two you land on still sound like close relatives of each other.
Same starting note, two complementary distances
Start from C and go up a semitone, and you land on Db - a minor 2nd. Start from that same C and go up eleven semitones instead, and you land on B - a major 7th. Same starting note both times; the two distances always add up to twelve semitones between them.
Why inverted intervals sound like relatives
This isn't a coincidence, and it isn't really about the two notes at all - it's about the wheel. An interval and its inversion always add up to a full octave, twelve semitones: 1 and 11, 2 and 10, 5 and 7. The wheel doesn't care which direction you travel around it, only how far - so the short way around is exactly the same distance for both members of the pair. That's the same interval class from Color Coding showing up again: a minor 2nd and a major 7th get the same color on the wheel because they're the same distance, just going the short way around versus the long way around.
The rest of the interval classes
The same pattern holds all the way through. Six of the seven interval classes pair up this way; the tritone doesn't - six semitones is exactly half of twelve, so subtracting it from twelve just gives you six back. It's its own inversion, with nowhere else to land.
Open and a little restless, but nowhere near as harsh.
Mellow and a bit wistful.
The two sweetest, most settled intervals in the system.
Clean, open, almost hollow - the least ambiguous sound there is.
Splits the octave exactly in half, so it has no partner - it inverts to itself.