I've put rotating kites in a single category, but in fact there are three quite distinct types:
The gyrokite is basically a gyroplane, that is, a helicopter with an unpowered rotor.
Some confusion exists as to how the gyroplane works, and indeed, how a helicopter can glide to a very safe and controlled landing after an engine failure. This becomes clear if you think of each rotor blade as a wing of a glider, which just happens to be going round in circles instead of in a straight line. Just like a glider, a rotor blade of a gyroplane generates lift as it slowly descends, and tends to accelerate if it's angled so as to descend faster. The gyrokite flies with its rotor tipped slightly away from the wind, so that the wind passes up through its plane, causing it to rotate and generate lift.
There is a fairly heavily marketed plastic gyrokite available. Those who've tried it generally seem to be of the opinion that although it makes a fascinating toy, it needs very smooth winds to fly successfully. Only buy one if you can afford the loss if it crashes and breaks! The main thing seems to be to make sure the rotor is spinning fast enough before releasing it. Here are some useful flight notes which were posted to rec.kites.
Many experiments with gyroplanes were performed, mainly by the Spaniard Juan de la Cierva (1895-1936), before rotary flight was conquered. In particular, it was found necessary to allow the blades to move up and down slightly as they rotate in order to allow for "dissymmetry of lift", or the fact that each blade flies alternately with a head wind and a tail wind as it rotates. It was also found necessary to allow each blade to swing slightly ahead of or behind its natural position. The lack of refinement of such details may account for the limited success of modern simple plastic gyrokites.
In the 30's, Captive Flight Devices of Pennsylvania marketed a gyrokite with a 36" rotor mounted on a machined shaft and a ball race. It also had wings and a tail, and flapping rotor blades. Needless to say, its weight meant that it needed a strong wind to fly, but it was reputedly very stable, as one might expect: the gyroscopic effect of the rotor should help to stabilise it, and its momentum should provide lift during lulls in the wind.
Hart gives a photo of a similar device in the Smithsonian, invented by Charles Chubb (or Chupp?), but no details are given in the text. The Chupp Roto-Flyer, thought to date from the 40's, has been studied by Gene Rock, who has put together detailed plans, and built a reproduction model, pictured here.
In addition, Daniel Bertolino has built several successful gyrokites. Here he is, flying one against a beautiful background in the French Auvergne, and here's a video clip of Daniel launching and flying it.
In 1943, the Germans deployed the Focke-Achgelis F.A.330 unpowered gyroplane for reconnaissance from submarines. It was towed by the submarine and lifted an observer to a height of several hundred feet. The rotor blades could be jettisoned "in an emergency", allowing the observer to descend by parachute. What sort of emergency was envisaged by the designers might not be immediately apparent, until you realise that the gyroplane would have been a give-away to the enemy if spotted - the strategy must have been "spot your target, then get down FAST!"
Whereas the gyrokite cannot fly without the rotor spinning quite fast because it uses the rotation to greatly increase the effective wind speed, rotation in windmill kites is more incidental. It adds greatly to the visual impact and may still contribute to the lift, though less so than in the gyrokite.
Sonja Graichen of Germany has been building award-winning kites at least since the mid 90's, but has recently gained international recognition for her highly original rotating creations. Working as an engineer, she found that her day-job failed to satisfy her considerable creative streak, which she then brought to bear on kite-building, with notable success. A web search on her name throws up a number of examples of her work, as well as some references to judo. A formidable lady! (I wonder if she found her skill at making people rotate in the air and land on the ground useful in creating kites which rotate in the air and stick to the sky?!)
Here, for example, is Sonja's Odysee, consisting of several counter-rotating wheels. Her Metamorphose uses four wind turbines to turn two rollers, driving a band of coloured film to produce continuously changing coloured patterns. A truly remarkable creation.
A horizontal spindle has two or more roughly semicircular paddles attached to it, as well as one or more circular or oval pieces mounted perpendicular to the spindle, for stability. Several different configurations along these lines are possible.
The original UFO kite was invented and patented by Ken Sams who often flew it in London at the Round Pond or from Westminster Bridge. He died in 2003 aged 80. Two paddles are formed from a single oval with the spindle running through its major axis. It has another oval of the same shape and size perpendicular to the spindle and at its centre. The paddles are often made of silvered mylar sheet decorated with holographic patterns, causing it to catch the sun and glint as it rotates. To onlookers who are unfamiliar with it, it can be hard to make out what it is, and hence its name!
The rotor kite is a little different. It has two disks perpendicular to the spindle near to its ends, and two or more paddles. The paddles are curved, causing it to rotate.
Both the UFO and the rotor kite work by the Magnus effect, which is the force that results from the interaction of an air stream and a rotating body. The curved flight of a spinning ball is due to the same effect. The direction of rotation of the kite is such that air passing over the top is flowing in the same direction as the paddles and so moves faster than the air passing beneath, flowing against them. Whenever the speed of an airflow increases, there is always a resultant drop in pressure, just as in the case of air flowing over an aeroplane wing, and this is what generates the lift.
In the case of the UFO kite, the paddles are flat so there is no obvious reason why it should spin. In fact, it tends to need a little help to get it started, but then continues to spin by a mechanism that is not altogether clear. The explanation would seem to be that air passing underneath moves between the paddles, helping them to rotate, whereas air passing over the top against their motion largely avoids them.
Here is a picture of a UFO made out of ripstop nylon and fibreglass rod by Anthony Thyssen, who has a discussion of rotor kites online which is well worth reading. He also has plans for a rotor kite - raw materials cost: virtually nil.
A number of variations have been tried, such as twisted winged box, and using a hexagonal or circular cross section instead of square, as in the "Revolver", shown here All have a tendency to drift to one side on account of the rotation, so it's usually best to fly them in counter-rotating pairs, as in this picture of two cylindrical ones by Anthony Thyssen. Sonja Graichen of Germany solved the problem by mounting two counter-rotating sets of twisted vanes (like the wings of a winged box, without the box) on a single spindle. A third fixed set of vanes at the front provides a convenient towing point.
Essentially a bol or basket consists of a parachute with an extra large vent, which may amount to as much as 90% of the diameter. It's made to rotate by secondary vents between the segments, arranged to direct air in a clockwise (or anticlockwise) direction. There do not appear to be hard and fast rules as to when a kite is a bol and when it's a basket, but those with a large main vent amounting to most of the diameter are generally regarded as bols, and the more parachute-like ones as baskets. Distinctions have also been drawn as to whether the secondary vents are adjacent to the leading (bridled) edge, or the trailing edge (the main vent).
Bols are sometimes confused with the circoflex, but this is in fact quite different. In particular, bols are typically made to rotate (though they needn't be), and they don't generate net lift. All the legs of the bridle are of equal length, and they meet at a point on the axis of the kite.
There are pictures of various bols and baskets on the Wind Climbers home page, which also contains plans for a 4ft basket, and a picture of Ed Hummel's 35ft diameter bol.