What you described is not an enharmonic relationship, but rather an inversion. Where one pitch is re-positioned an octave above or below the other pitch.

The inversion of a minor third is a major sixth.

Fifths invert to fourths.

Seconds invert to sevenths.

Octaves invert to unisons.

Enharmonic relationships are when different "spellings" are used for the same thing. Intervals and pitches can both exhibit enharmonic relationships.

For an interval let's consider 3 half-steps. We could spell that as C to E flat (a minor third) or we could spell it as C to D sharp (an augmented second.) Different names and spellings for what are enharmonically the same interval distance of 3 half-steps.

For a pitch example let's look at the E flat and D sharp of the previous example. They are both the same pitch (same key on the piano is another way to look at it) but different spellings are used.

What you described is not an enharmonic relationship, but rather an inversion. Where one pitch is re-positioned an octave above or below the other pitch.

The inversion of a minor third is a major sixth.

These are the basic interval inversions:

• Seconds invert to sevenths.
• Thirds invert to sixths.
• Fourths invert to fifths.
• Octaves invert to unisons.

Your original question could be re-worded as...

How to determine if an interval is [an inversion] or not?

There is nothing inherent in an interval to say it is an inversion. So, a minor third is an inversion of a major sixth, but that should not be misunderstood to mean minor thirds are inversions or minor thirds are created by inverting major sixths.

Usually something will be labelled an inversion from some initial reference point. An example that comes to mind is invertible counterpoint. This where we have two melodies in counterpoint and the lower one is raised an octave to place it above the other melody. When that is done all of the intervallic relationships become inverted. So intervals that were thirds in the original counterpoint will become sixths in the inverted counterpoint. In that case we could speak of the sixths as inversions of the thirds. There are other inversion relationship, invertible counterpoint is only example.

Enharmonic relationships - a completely different matter - are when different "spellings" are used for the same thing. Intervals and pitches can both exhibit enharmonic relationships.

For an interval let's consider 3 half-steps. We could spell that as C to E flat (a minor third) or we could spell it as C to D sharp (an augmented second.) Different names and spellings for what are enharmonically the same interval distance of 3 half-steps.

For a pitch example let's look at the E flat and D sharp of the previous example. They are both the same pitch (same key on the piano is another way to look at it) but different spellings are used.