The Diminished Seventh chord
The dimished seventh chord is formed entirely from intervals of a minor third, stacked on top of each other; it's named after the interval between the root note and the 'top' note.
The diminished seventh rooted on A, for example, is made up of A C Eb Gb. Like the augmented triad, it is symmetric; the dim7 chords rooted on C, Eb and Gb each contain the same notes as A dim7.
In classical theory, the diminished seventh is a cadential chord: it has a strong 'pull' to chords rooted a semitone above, and so lends momentum to a cadence. Perfect cadences preceded by a diminished chord can be very strong and final-sounding. (And very J.S. Bach, as well; any A level students who have to harmonise Bach chorales would do well to take note.)
Alternatively, the diminished seventh rooted a semitone below the tonic chord (and resolving to it) makes a satisfying perfect cadence; such a chord shares three notes with the dominant seventh it replaces.