Negative interval
A negative interval is an interval that goes down the scale but up in pitch, and vice versa. For example, in just intonation the Pythagorean comma is an ascending interval, and C♯ is higher than D♭. (Uninflected note names are here assumed to refer to 3-limit JI.) But because this comma is an augmented unison minus a minor 2nd, it can't be described as a unison or a 2nd. Just as a 5th minus a 2nd is a 4th and a 4th minus a 2nd is a 3rd, a unison minus a 2nd must be a negative 2nd.
The interval between C♯ and D♭ (or equivalently between D♭ and C♯) is a negative diminished 2nd. We say "equivalently" because the interval between two notes is a vertical or harmonic interval, whereas the interval from one note to another is a horizontal or melodic interval[note 1].
"Negative" does not mean "descending". The melodic interval from D♭ to C♯ is negative but not descending. A melodic interval can be descending but not negative. For example, the melodic interval from D down to C is a descending major 2nd. Furthermore an interval can be both descending and negative. For example, the melodic interval from C♯ down to D♭ is a descending negative diminished 2nd.
"Negative" does not mean "inverted". The inversion of a diminished 2nd is an augmented 7th. The inversion of a negative diminished 2nd is a diminished 9th.
Temperaments
In certain temperaments such as meantone, the fifth is flattened sufficiently such that the Pythagorean comma becomes descending. It's no longer negative, and is simply a descending diminished 2nd. However, negative 2nds do occur in meantone. (In fact, multiple negative 2nds, 3rds, etc. inevitably occur in every tuning of rank-2 or higher. We can simply repeatedly diminish a 2nd or a 3rd until it becomes descending, then flip it to make it ascending.) In the case of meantone, the kleisma (fifthspan of +19) is a negative 2nd.
Interval arithmetic
Adding or subtracting a negative interval is the same as subtracting or adding the corresponding positive interval.
For example, what is an octave plus a Pythagorean comma? We must subtract a diminished 2nd from an octave. We know that P8 - m2 = M7. If we diminish what we're subtracting (m2), we will augment the result. Thus P8 - d2 = A7, an augmented 7th, e.g. C-B♯. Likewise a major 3rd minus a Pythagorean comma is a diminished 4th, e.g. C-F♭. An extreme example: the sum of two Pythagorean commas is a negative triply-diminished 3rd, e.g. C-A♯♯♯.
Prevalence in just intonation
Within a single piece of music, it's quite rare to find two notes a Pythagorean comma apart. Thus negative 2nds are relatively unimportant in 3-limit JI. In 5-limit JI, the simplest (i.e. least odd-limit) negative 2nd is the schisma = [-15 8 1⟩ = 2¢, also rare.
But in other tunings negative 2nds are commonplace. For example, in 7-limit JI, the interval from 7/5 (a diminished 5th) up to 10/7 (an augmented 4th) is 50/49 = 35¢, a negative diminished 2nd. Furthermore, the interval from 16/15 (a minor 2nd) up to 15/14 (an augmented unison) is 225/224 = 8¢, another negative diminished 2nd.
Negative minor 2nds are possible but rare. For example, 1728/1715 = [6 3 -1 -3⟩ = 13¢ is equal to (8/5)/(7/6)3, a minor 6th minus three minor 3rds.
Reentrant scales
A reentrant scale has at least one negative interval, going backwards relative to the general direction of the scale. Although a reentrant scale is not strictly ascending or descending, its ascending and descending forms are determined by its general direction.
Reentrant scales are mostly relevant when applying extreme tunings to abstract scales, causing some steps to have a negative size in order to preserve the abstract scale's usual structure. For example, if you try to generate a MOS scale with a generator whose size falls outside of the generator range of all possible MOS patterns with the same given number of notes, you will obtain a MOS scale with a negative step ratio. A 7-tone scale with a 295 ¢ generator is just outside of the range for 4L 3s, and can be interpreted as a 4L 3s scale with 315 ¢ large steps and -20 ¢ (negative) small steps, whereas considering the scale's pitches in ascending order leads to a ternary, MV4 scale interpretation.