In my other articles, I mentioned how Intervals and Chords are inverted.

For an interval, it really simply means putting the lowest note above the higher note (move it up an octave) or vice versa.

Chords are actually a little more funky in a way: IF you stack the chord as "small" as you can, then putting the lowest note above highest note is similar to what you do with intervals (which is probably why we call it inverting) but the result is somewhat different. What we're really doing is "rotating" a set of notes where each element in turn can be the lowest note. And ultimately, that's all we care about - whether the Root, the 3rd, or the 5th, etc. is the lowest note, and not how the remaining notes are distributed.

But the word Inversion has two other common usages in music!

In one, Inversion, means to "turn upside down" or a better word, "mirror" a musical line.

For example, if we invert:

C D E C, which is up a step, up a step, down a third (+1, +1, -3*)

The inversion would be -1, -1, +3, so:
C B A C

This can be done two ways, called "Tonal" and "Real" (or Exact, or Literal). A Tonal Inversion stays in the key.
C D E C inverts to C B A C in the key of C Major. In the key of F major,
C D E C would invert to C Bb A C.

A Real Inversion maintains EXACT interval distances, so:
C D E C - up a M2 up a M2, down a M3
inverts to
C Bb Ab C - down a M2, down a M2, up a M3

*IN 12 tone theory, we would say up a M2 is +2 not +1, I just did it by letter name here for convenience.

Most commonly, Inverting in this sense is encountered when motives in contrapuntal pieces or subjects in Fugues are inverted. Additionally, it is one of the mainstays of operations one can perform on a 12-tone row. However it is quite common throughout history. The famous opening motive from Beethoven's 5th Symphony is treated in Inversion in the bass parts when the melody first enters in the upper parts (and this inversion takes the interval of the 3rd of the original, and treats it as a 4th in this inverted form, so we call it "varied" or "altered", etc.).

The remaining common use of the word Inversion is in relation to Invertible Counterpoint. This is more like the "flipping" encountered in Interval Inversion except rather than just a pair of notes, it's an entire musical line. Here are two lines:

C D E C F E D-
C B C E D C B-

If I invert these two lines, it yields:
C B C E D C B-
C D E C F E D-

Musically speaking, there are a couple of ways you could do this:

1. One of the parts could jump an octave (or multiple octaves) and the other remain stationary (at the same original pitch). For instance, in the first example above, the lower line could be Middle C (C4) and the upper C5 (an octave above middle C). In the second example, the upper line - the "original" could still be at C4, but the lower line would now be in the Bass clef on C3 (an octave below Middle C).

2. "Swap" the lines. For instance, in the first example the lower line could be C4 and the upper C5, and then in the second example, the lower line would be C4 and the upper line C5. In a sense, each melody has swapped its octave with the other melody (but the same player would play the lower one both times, etc.).

3. It is possible with multiple lines (3+ parts) to invert them more times, and this is much like the way when invert chords, except that in this case we just say the "passage" is in Invertible counterpoint. In such an instance, two of three lines could change positions, and we'd say those two have "inverted" though we'd probably still say the whole passage of three parts is Invertible Counterpoint even though one of them didn't do much. With 3 lines, there are six possible arrangements (you see, the same is true of chords, but we only talk about the three inversions - one instance for each note). Composers don't typically use all possible arrangements but any passage that inverts two or more of the parts we'd call Invertible Counterpoint (or mention the specific parts that have been inverted, like, Duple Invertible Counterpoint between Soprano and Tenor, etc.). For a wonderful example of 5-part Invertible Counterpoint check out the Finale of Mozart's "Jupiter" Symphony in the Fugato section.

So when someone says "X line is an inversion of Y line", make sure you understand whether they're talking about a "mirror" inversion, or a "swapping" of parts.

And try inverting your own music - for example, don't always put your melody in the soprano - see if it will work in the middle, or lower voices. and some of those parts can be move around. If you want to keep consistency, but add a bit of variety, inverting a pair of lines can do that simply. Mirror inverting can do similarly, but has the added advantage of there being a little "secret" there - everyone might not notice that line X is a mirror of line Y! But somehow, subconsciously, it seems "familiar" to them. Try mirroring chord progressions too - I - iii - V can be mirrored by I - vi - IV for some interesting results. You can also invert the rhythm of one part with the rhythm of another, but leave the pitches as original between two lines (swap the rhythm but not the pitches) for interesting results.

A final caution - we usually don't say a line is inverted, or Invertible Counterpoint is being used when a SINGLE line appears in various voices - like if it starts in the Bass and then goes to the Soprano - only if it swaps with another part (lower part goes above the higher or vice versa) is it an inversion.

fine.