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.