The timepiece comprises a timekeeping mechanism and a switching device arranged to switch this timekeeping mechanism between two states. The switching device comprises a movable switching organ and a rotary control organ arranged to be stepwise driven in order to successively occupy a plurality of distinct angular positions about its axis of rotation. The movable switching organ and the rotary control organ respectively support a first magnetic structure and a second magnetic structure arranged so that, in a first angular position of the rotary control organ, a first magnetic force acts on the switching organ in a given direction, and so that, in a second angular position of the rotary control organ, a second magnetic force acts on the switching organ in the opposite direction.