The #1 rule of sports biomechanics is the conservation of angular momentum. It dictates that whenever an athlete performs an acrobatic jump, the angular momentum that he has created on takeoff is to stay unchanged until he lands. He can control the speed of rotation by expanding or retracting his limbs, but he can't just randomly stop rotating in mid-air and then continue again out of nowhere.
For rotations around multiple axes (twists etc), I understand that the conservation of momentum should work for each of individual axes.
Now take a look at this jump (starts at 0.51): https://youtu.be/sb82tVOq2dY?t=51
On takeoff, the diver initiates a flip with a twist (a spin around both vertical and horizontal axes at the same time). But then, in the middle of the jump, he somehow kills the vertical component of rotation and converts to a plain frontflip.
In another video, you can see the opposite: https://www.youtube.com/watch?v=fwDGrNKiTi8
Here, the athlete initiates a pure frontflip rotation on takeoff. However, before the last flip, he somehow initiates an additional rotation around vertical axes, pulling that 180 in the end seemingly out of nowhere. And I've seen people pulling even 360s like that out of nowhere.
So what's going on there? Is it possible for an athlete to initiate or kill angular momentum in mid-air somehow? Or is there some other effect at play?
Best Answer
To explain how orientation can change whilst angular momentum is conserved it is first best to look at a slightly simpler system - a cat in free fall!
Here is a series of photographs taken in $1894$ which shows a cat turning its body to ensure that it lands on its feet.
This gif file illustrates how a cat changes its shape to rotate and yet still to conserve angular momentum.
Finally here is a video of such an event with the cat suffering no harm.
So the key is changing body shape to achieve a rotation whilst conserving angular momentum.
This is shown using a selection of stills from the gymnast video.
First head on.
Arm movements starting in slide $\rm d$ initiate the twisting of the gymnast.
From the side.
Here is a dive executed in the video referenced by the OP.
The diver when on the diving board cannot use it to start a twisting rotation as that rotation could not be removed towards the end of the dive and I think that is also against completion rules.
The Physics of somersaulting and tumbling is explained in an article published in Scientific American.
By moving the arms a diver can start and stop a twist.
The somersault rotation continues from start to finish but before entering the water the diver increases the moment of inertia about a horizontal axis by stretching out thus reducing the speed of rotation.
By timing the entry to perfection and whilst still rotating the diver enter the water with the smallest horizontal profile.
Note the rotation continuing under the water.
This slow motion video of a twisting somersault shows clearly how the arms are used to initiate twisting.