Dear Daisy,
You are correct in your description of the diminished chord's properties (one/minor third/flatted fifth/double flat seventh or sixth).
When you both diminish (shrink) and augment (stretch) a chord, it's called a "demented" chord. (Just kidding)
I have also found a use for diminished chords as substitute chords (in blues/swing).
On the full sized keyboard, if you were playing in the key of "C" and at the end of a blues progression in the "turnaround" you may encounter what I call a "3-6, 2-5, 1" (Em-A, Dm-G, C).
If you play the proper bass notes (or walk them) with the left hand(E, A, D, G, C) you can substitute some of the chords (R.H.) which finger nicely. (Em-Edim (instead of A), Dm-Ddim (instead of G), C79).
Obviously this doesn't work in all styles of music, but it does have occasional applications as the earlier posters stated.
Perhaps you can come up with a relationship formula which will tell you where to substitute and then let the ear be the final judge.
By the way, they say you can't invert a diminished chord. Who the hey do they think they are. If you do consider them invertable, that means there are only 4 or 5 of them.
Happy diminshing!
mav out