How often the sense of a piano work is contained in the left hand, so often neglected because it does not have the melody.
It took me a while to figure out that the usual division of music into three elements, melody, harmony, and rhythm, was entirely phoney, that a lot of music is a magical combination of the three so that they are in fact inseparable.
Having been trained as an organist I spent a lot of time identifying and articulating fugue subjects and counter-subjects. One day I woke up to how much more interesting fugues are if you simply relax and listen. What you hear tends to be a fascinating dispersal of long notes amid running passages. The trajectory of the long notes is utterly compelling.
I recommend to people that they approach their music via the long notes, not via the themes or the melodies. The long notes convey both harmony and rhythm, and sometimes (especially in Beethoven!) even melody.