Only four days a year does the sun culminate exactly at twelve on a conventional watch: April 15, June 13, September 1, and December 25. On the other 361 days there is a difference, and the equation of time is the complication that makes it readable on the dial.
The cause is twofold. Earth's orbit is elliptical, not circular: the Earth accelerates near perihelion and slows at aphelion, changing the apparent speed of the sun in the sky. To this is added the inclination of Earth's axis of about 23.5°, which further alters the time of solar culmination throughout the months. The result is a difference that oscillates between −14 and +16 minutes over the course of the year, with extreme values reached around February 12 and November 3.
The classical mechanism is based on a cam with a kidney-shaped profile, engraved or cut into a disc that completes a full annual rotation. A follower traces the cam's profile and transmits the movement, through a rod, to the equation hand. The shape of the cam physically encodes the curve of the equation of time, which astronomers have calculated for centuries and which does not change appreciably on a human timescale.
There are two families of displays. In the static, or retrograde, version, a hand on a graduated scale shows the value in minutes of advance or delay relative to civil time: useful for knowing when to correct a sundial, less immediate in daily reading. In the marching version, introduced by Breguet in its reference 5887, a second minute hand—often ending with a golden sun—advances or retreats in real time relative to the civil hand. The two hands move apart and come together throughout the year: there is no calculation to do, you read the true solar time directly as you read the time. It is the mechanically most complex version, because the marching hand must move continuously, not jump at intervals.
In some astronomical watches, such as certain Vacheron Constantin or the Solaria by Ochs und Junior, the equation of time integrates with indications of sunrise, sunset, solar altitude and declination, calibrated to a specific latitude. In these cases the cam alone is not enough: differentials and additional modules come into play. At the bench I have seen complications of this type with transparent sapphire cams, legible from the caseback: a technical and visual choice that I do not find trivial.
I do not know the precise position of the cams in the movements of each manufacturer, and I do not report numbers that I have not verified directly on the original technical documentation.