In general as well as more specifically in chess.
It seems some people use "turno", "jugada" and "movimiento" interchangeably, but is there any technical difference (like one is a single play and another is a full turn)?
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Movimiento is the most general of the three words you are comparing. You can understand it as any intentional and organic change, generally strategic, in the development of any activity. It is not necessarily from the scope of games or sports. The chess player makes "movimientos" (or "movidas"), but so do the military strategist, the broker and the politician.
A jugada, on the other hand, could be understood as a movimiento in the scope of a sport or game (although of course you can use it metaphorically in phrases like "el presidente hizo una buena jugada al despedir a sus asesores").
A turno occurs in activities in which the participants make their movimientos or jugadas in a certain period of time, which is successive and has been assigned according to rules. In chess each turno is a jugada, but there are other games, such as soccer, in which the jugadas are not taken in turnos.
Movimiento = movement, from the verb "mover" (to move). It means that you perform any change on your desk.
Turno = turn, it's the time when you can do a "movimiento", but you can also skip your turn and let the other player go ahead.
Jugada, altough it can be a synonym of "movimiento", I don't see them to be equal. As I see it, "jugada" is "an intelligent plan to achieve your goals". It can have more than one step, so it could be more than one "movimiento"
In sum: Turno - time Movimiento - any kind of movement Jugada - one or more "movimientos" that have been studied in order o achieve something.