A diabolo looks like a toy that should need a track. Two hollow cups joined by a narrow waist, a loop of string between two handsticks, and the whole thing sits in the middle of that loop apparently resting on nothing. Then it is tossed two metres into the air, turns over, and comes down still level enough for the string to catch it. Nothing is supporting it in flight. The string is not a rail; it is a drive belt.
The trick is that the string supplies rotation, not lift. Pull one hand up while the other gives slack and a short length of cord drags across the axle. Friction turns that drag into torque. Repeat a few times, alternately, and the diabolo is spinning fast enough that it no longer cares which way it is pointing. Once that happens the string can be slack, then removed altogether, and the toy keeps its attitude while it flies.
Why the cups are at the ends
The shape is doing more than looking decorative. Almost all the mass sits in the two rims, as far from the axis as the mould allows, with very little at the waist. For a given spin rate that arrangement gives a large moment of inertia, and a large moment of inertia means a large angular momentum. Angular momentum resists any attempt to change the direction of the spin axis. Tilt the diabolo as it leaves the string and it does not tumble end over end the way a stick would; it wobbles slightly around a direction it mostly keeps.
A solid cylinder of the same weight would be much easier to knock off axis, because more of its mass would sit close to the centre where it contributes little. Spinning tops solve the same problem with a different trick: a sharp point and a low centre of gravity. A diabolo has no point and no floor, so it buys all of its stability from the distribution of its own mass. The same reasoning explains why a three-bladed fan is balanced the way it is: what matters is not how much material you use, but where you put it.
Why the string stays in the middle
Watch the cord closely while it is driving. It does not wander along the axle. The two inner faces of the cups slope down towards the waist, so a cord that drifts to one side is walking uphill and tends to slide back to the lowest point. That self-centring is why the waist is concave rather than a plain straight rod: a flat axle would let the string creep across, rub on a rim, and eventually flip the toy out of the loop. The slope is a guide with no moving parts.
The same geometry sets a limit on how hard you can pull. Too much tension and the cord grips rather than slides, the torque spike tips the diabolo, and it climbs the string. Skilled players work with a short, light contact repeated often, not one long hard pull. That is also why the cord is cotton or a soft synthetic rather than a slick fishing line: a little friction is the whole mechanism, and polishing it away would leave a toy that cannot be started.
Single-cup versions make the point by breaking the symmetry. With mass at one end only, the spin axis is not the same as the balance axis, and the toy wants to precess in a wide circle. Players compensate by holding the sticks at an angle and continuously correcting. It is a harder object to learn precisely because the neat coincidence between where it balances and where it spins has been removed.
What the string taught the toy makers
Once the fall is predictable, everything else can be built on top of it. Tosses, catches behind the back, passing between two players, and the long whistling note from the vented cups all rely on one fact: the object returns to a known orientation without being aimed. That is the same bargain a kite makes with the wind, and the reason children's games change so slowly — a shape that solves its problem this completely has very little left to improve.
You can check the whole argument in a minute. Spin it up, then let the cord go completely slack and watch it hang in the air for a moment. It does not drop while it is spinning fast; it simply holds its angle. Now stop it with a gloved hand and throw it again. Without the spin the same object tumbles instantly. The cups, the cord and the string tension were all unchanged. Only the angular momentum was removed, and with it the entire trick.