Mirror, mirror on the wall, which is the most perfect one of all.
The principle of mirror intervals is this: create intervals starting from middle C going up, then make those same intervals going down from middle C. Use only the white keys and notate them. Then compare the size by counting the number of all the keys skipped. This means you are comparing up and down in the key of C, starting with what call the “tonic”.
If the distance is the same both ways, the interval is perfect. When it is different, it is major going up and minor going down.
This little trick only works when you start with the tonic of a major key. You can do the same thing with the other 11 major keys, but it’s more complicated to see the results.
The matter of interval complements is a slightly different subject. Interval complements always add up to eight, which happens because one note or key is always counted twice when intervals are added.
When you add intervals, you always lose something because intervals count your starting and ending point. It’s a bit like taking two steps but counting them as three because you count where you start and then count your last step. Or you take four steps and count them as five, for the same reason. This creates musical math: 3+3=5, 3+3+3=7, and so on. So three 3rds = a 7th. CE, EG and GB are all 3rds. Those three 3rds, stacked, add up to a 7th.