Of these 12 different intervals only one is its own reciprocal: the Diminished 5th, also called the Augmented 4th. Moving up or down by this interval brings us to the same place (if you ignore the change of octave). This property isn't just a neat mathematical trick but has some wonderful effects when introduced into a harmony.

Another way to look at the Diminished 5th is as the "farthest poles" available for two notes to lie upon. The Diminished 5th interval extends to exact opposite sides of the Circle of Fifths, and it also extends diametrically across the Circle of Semitones (C-C#-D-Eb-E-F-F#-G-Ab-A-Bb-B-C).

When two notes a Diminished 5th apart are rung together it sets up a harmonic tension which is called a tritone. "If there is a bright center to the Universe you are on the planet that it is farthest from." The tritone is the place where everything culminates - there is no place to go but back.

The most familiar example of a tritone shows up in the V7 chord. In the key of C this is G7. Playing the G7 chord sets up a tritone which wants to immediately resolve to C, down a 5th. In fact, the movement to the 5th below is the strongest leading tendency of the tritone.

The G Dominant 7 chord has the most power to lead because it contains not just the tritone but three other tones with tendencies to resolve to C: the Dominant 7 or "leading tone" (B), the Fourth (F), and the Fifth (G).

The Leading Note (B)
The 7th note of the C major scale, the leading note B, tends to resolve naturally to C (the tonic). B is the 3rd of G7, and the movement from B to C can be heard in the V-I resolution, when the 3rd of the dominant chord of G major (B) resolves to the root note of the tonic, C major.

Fourth Note (F)
The 4th (F) has a strong tendency to resolve harmonically. When included as a 7th in the dominant chord, G7, it moves to E, the 3rd of the scale and the tonic chord. Try playing C major with F (Csus) and move F down to E to hear how this movement works with the note C.

Fifth Note (G)
The movement B to C can be heard using a two note V chord. With G this resolves C major. The bass (G) moves naturally up a fourth (down a 5th) - G to C.

Tritone
The combination of F and B sets up a strong tension that needs to resolve. The tritone interval is the same when F is placed above B. Each note must still move by a semitone for resolution.

Every Tritone Is Really Two Tritones
Depending on how you play the notes of a tritone it can have two different ways of resolving. F below B is an Augmented 4th tritone, the kind that likes to resolve to C. F above B has a tritone interval of a Diminished 5th, and this kind prefers to resolve to Gb/F#. In this case the two tones simply resolve in opposite directions. Instead of F-E and B-C the movement is F-F# and B-Bb.

The resolution to Gb/F# can be used with F below B, and likewise the movement to C can be used with F over B. It all depends on the context - and your ear.