Contrapunctus XII from the Art of the Fugue (Die Kunst der fuge BWV 1080) is the first of two mirror fugues. In these fugues each voice of the fugue called rectus is inverted and used to create a second fugue called inversus. The voices from the rectus fugue are inverted into the second fugue in the following order:

Inversus						Rectus
	soprano				soprano
		alto		alto
		tenor		tenor
	bass				bass
Rectus						Inversus

In previous fugues Bach used the inverted subject of the Art of the Fugue. In these fugues he creates a fugue that can be inverted from beginning to end.

In the following animation you can see the first measures of the rectus fugue. The inversus fugue looks like a reflection on a mirror of the rectus fugue:

The tenor (rectus) and alto (inversus) follows with the first answer:

Then the alto (rectus) and tenor (inversus) in measure 10 with the second subject:

Finally the soprano (rectus) and bass (inversus) end the exposition:

In the inversion process Bach also needs to invert the modulations. In the rectus version the answer takes us to the dominant key (A minor, note the G sharp and B natural in measure 15). In the inversus this modulation goes to the subdominant key (G minor, note the F sharp in the soprano and the E flat in the alto on measure 15). Similarly, a visit to the relative major on the rectus (F major, last measure of the previous example), translates into the submediant in the inversus (Bb major in first inversion):

r e c t u s			mediant F major III
		dominant A minor v
	tonic D minor i
i n v e r s u s		subdominant G minor iv
			submediant Bb major VI

This harmonic dimension presents new difficulties when we want to create an invertible fugue. In this case we reach the mediant (F major) in measures 14 and 15 using its VII degree (vii^4/3/III - III6) while the submediant is preceded by the dominant chord of D minor using a deceptive cadence (V⁷ - VI).

A countersubject created using the last notes of the subject in the inverted and original form appears with each voice: