The clutch is the critical point of the entire chronograph architecture: it determines whether the seconds hand starts cleanly or with a jerk, whether the balance wheel loses amplitude at engagement, whether the wheels wear prematurely. There are two main geometries, and the choice between them is functional, not aesthetic.
The horizontal clutch, also called lateral, is the historical solution. The chronograph wheel engages radially on the command wheel: when you press the button, the two wheels touch sideways. It is compact, relatively simple to build and service, which is why it has survived in important calibers like the Zenith El Primero. The drawback is well-known: at the moment of engagement, the seconds hand may make a small irregular jerk, and there is always a transfer of lateral force that marks the teeth of the wheels over time.
The vertical clutch works concentrically on the same axis. When the chronograph is stopped, the wheels are disconnected axially; at startup, the mechanism descends and the surfaces couple through friction. The result is that the hand starts without jitter, balance wheel amplitude loss is minimal, and wear on the flanks of the teeth is almost absent. Tradition attributes the industrial adoption of this system to Seiko with the 6139 caliber in 1969; today it is found in calibers like Tudor's MT5813 and Carrera's TH20-00. Technically superior, but more delicate to disassemble and adjust.
Separate from the engagement geometry is the column wheel, often cited alongside the vertical clutch as if they were the same thing. They are not. The column wheel is the command device that marks the positions for start, stop, and reset; it can work with either a vertical or horizontal clutch. Its alternative is the cam mechanism, more economical to produce and less refined to the touch.
There is also the horizontal clutch used in reverse, exploited to reduce movement thickness: the Angelus A-500 caliber is a recent example, where the flat geometry is a deliberate construction choice.
A separate category is the safety clutch of the NOMOS DUW 6101: here the clutch does not control the chronograph but protects the date mechanism from the so-called danger zone, making a wheel slip under friction instead of transmitting force to the date wheel. Same logic of disengagement, completely different application.
In the domain of resonance, Armin Strom uses the term clutch for a patented solid steel spring that connects two independent balance wheels in the ARF21 caliber. It is not a clutch in the traditional sense: it is an elastic coupler that transfers impulses between the two oscillators until they come into phase. The name is used broadly, but the function is distinct.