Timepiece comprising: - a timekeeping mechanism able to switch between a first determined state and a second determined state; - a switching device (1; 301) arranged to switch the timekeeping mechanism between said first state and said second state, this switching device comprising a movable switching organ (23; 223; 323a, 323b) and a rotary control organ (21; 121; 221), the rotary control organ being arranged to be stepwise driven in a given direction of rotation in order to successively occupy a plurality of distinct angular positions about an axis of rotation (22; 350), the switching device being arranged so that a stepwise rotation of the rotary control organ (21; 121; 221) in said direction of rotation causes a reciprocating movement of the movable switching organ (23; 223; 323a, 323b) substantially in a plane perpendicular to said axis of rotation between two stable radial positions for at least one part of this movable switching organ, said organ being transferred from a first to a second of the two stable radial positions, in order to trigger a first switching of the timekeeping mechanism, when the rotary control organ is placed in a first angular position from among the plurality of distinct angular positions, and the movable switching organ being returned to the first of the two stable radial positions, in order to trigger a second switching of the timekeeping mechanism, when the rotary control organ is placed in a second angular position from among the plurality of distinct angular positions, characterised in that the movable switching organ (23; 223; 323a, 323b) and the rotary control organ (21; 121; 221) respectively support a first magnetic structure (31; 131a, 131b; 331a, 331b) and a second magnetic structure (33; 133a, 133b; 333) arranged in order to have a mutual magnetic interaction that allows the timekeeping mechanism to be switched between said first and second states on command, one of the first and second magnetic structures comprising at least one first magnetic pole and the other of the two magnetic structures comprising at least one second magnetic pole and a third magnetic pole with opposite polarities and being able to successively interact with the first magnetic pole, the first and second magnetic structures being arranged so that, in the first angular position of the rotary control organ, a first magnetic force, which is generated by a magnetic interaction between the first and second magnetic poles, acts on the switching organ in order to transfer said organ to the second of said two stable radial positions and so that, in the second angular position of the rotary control organ, a second magnetic force, which is generated by a magnetic interaction between the first and third magnetic poles and is in the opposite direction to the first magnetic force, acts on the switching organ in order to return said organ to the first of said two stable radial positions. - a timekeeping mechanism able to switch between a first determined state and a second determined state; - a switching device (1; 301) arranged to switch the timekeeping mechanism between said first state and said second state, this switching device comprising a movable switching organ (23; 223; 323a, 323b) and a rotary control organ (21; 121; 221), the rotary control organ being arranged to be stepwise driven in a given direction of rotation in order to successively occupy a plurality of distinct angular positions about an axis of rotation (22; 350), the switching device being arranged so that a stepwise rotation of the rotary control organ (21; 121; 221) in said direction of rotation causes a reciprocating movement of the movable switching organ (23; 223; 323a, 323b) substantially in a plane perpendicular to said axis of rotation between two stable radial positions for at least one part of this movable switching organ, said organ being transferred from a first to a second of the two stable radial positions, in order to trigger a first switching of the timekeeping mechanism, when the rotary control organ is placed in a first angular position from among the plurality of distinct angular positions, and the movable switching organ being returned to the first of the two stable radial positions, in order to trigger a second switching of the timekeeping mechanism, when the rotary control organ is placed in a second angular position from among the plurality of distinct angular positions,