Mnemonics, and an honest gap
Going up: The Entertainer. After the little chromatic run-up, Joplin's melody leaps a minor 6th, E up to C, and that leap is the tune's signature. For something newer, the string hook in Call Me Maybe outlines the same interval.
Going down: this is where the songbook thins out. Descending minor 6ths open very few melodies anyone reliably knows, so rather than hand you an obscure show tune, we'd point you at the inversion trick below and at practice itself; the descending button up top exists precisely because no song will carry this one for you.
The jazz songbook does carry it, if you know the records. Mark Levine's interval table fixes the ascending minor 6th to Woody Shaw's "In A Capricornian Way" and the descending one to Joe Henderson's "Serenity."
On the staff
The dark consonance
The minor 6th is a consonance, filed with the thirds and the major 6th in the warm, blending family, but it's the member with the shadow on it. In a minor key it's the distance from the tonic up to le, the lowered sixth degree that gives minor much of its color. Invert it and the shadow lifts: the same two pitch classes flipped become a major 3rd, the brightest interval in the family. Few inversion pairs are as instructive to hear back to back.
That inversion is also the practical handle. The wide quality discriminations (minor versus major 6th) are harder than the narrow ones (minor versus major 3rd), so when a sixth goes by unnamed, drop its top note an octave in your head and read the third that remains. Bright leftover, minor 6th. Dark leftover, major 6th. It feels like cheating; it's just theory doing its job.
What it gets confused with
The major 6th above it, for the reasons just covered. The perfect 5th below it is the other near miss: one semitone apart, but the 5th is hollow and resolved where the minor 6th leans with feeling. If the interval sounds like it's expressing something, it's not the 5th. The minor 7th, two semitones wider, has left consonance behind entirely and wants to resolve, which the 6th never does.
How Aubel handles it
The minor 6th unlocks late, seventh beyond the starting set, after both thirds and its major sibling are settled. By then the inversion trick has something to grab: the major 3rd it flips into is one of the best-trained sounds in your pool. The detail screen leads with The Entertainer.