Actually the inversion of a third will always be a sixth - not a third as posited in the title of your question. 3 + 6 = 9.
The inversion of a fifth will be a fourth 5 + 4 = 9. The inversion of a fourth will be a fifth 4 + 5 = 9 and so on.
An inversion of an interval (by definition) is simply flipping (inverting) the two notes comprising the interval so that the note on top is now on bottom and vice versa. It amounts to turning the interval upside down (inverting it).
One of the notes will always move by an octave which results in the two intervals adding up to contain 9 notes because each interval (the original and inversion of the original) contains both original notes (resulting in them being counted twice - once in each of two interval depictions).
In a true inversion, the notes will always be the same - only the order will change.
The picture below shows how to create an inversion of an interval by moving one of the notes up an octave.
Similarly you can invert the opposite way by moving the highest note an octave lower. But the two notes will be the same and the interval value will change so that the two will add up to nine.
It is certainly valid to define a descending third as an interval of a third and an ascending third an interval of a third. They are just not technically considered true "inversions" in accordance with the understanding of that term in music teaching.