What's the physics behind the jumps with seemingly non-conserved angular momentum? 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?
 A: A spinning object in free space (which the diver is a good approximation of) without any external forces acting on it, will keep the same angular momentum. However, note the difference between angular momentum and angular velocity. The former is conserved but the latter may not be, necessarily. Everyone knows of the basic example where a figure skater pulls in her arms to spin faster. This is a case where the angular velocity vector increases in magnitude but the direction of the angular velocity stays the same. However, it's also possible for the direction of the angular velocity vector to change. In other words, it's possible for the rotation axis to change.
Formally, to understand this, one can consider the full vector form of angular momentum for a collection of point masses, assuming they are rigidly connected, and also assuming the origin is the center of mass. This is:
$$
\mathbf {L} =\sum _{i}\mathbf {r} _{i}\times m_{i}\mathbf {v} _{i}
$$
Where $\mathbf{L}$ is angular momentum, $\mathbf{r}_i$ is the position of each particle w.r.t. the origin, and $\mathbf{v}_i$ is the velocity of each particle.
Imagine a simplified figure skater where there are just two particles -- the figure skater's arms (or hands), each of length 1, with the rest of her body being massless for now. Also assume we are in a coordinate system where the x direction points outwards from her body. Thus at time 0, without loss of generality we can say $\mathbf{r}_0 = \{1, 0, 0\}$ and $\mathbf{r}_1 = \{-1, 0, 0\}$. If $\mathbf{r}_i$ decreases (while still keeping the same direction), then $\mathbf{v}_i$ must increase (while still keeping the same direction) to keep $\mathbf{L}$ conserved. However, if the direction of $\mathbf{r}_i$ changes (say, she keeps one arm horizontal but moves the other arm up at a 45 degree angle), so then we might have $\mathbf{r}_0 = \{1, 0, 1\}$ then the direction of $\mathbf{v}_0$ will also have to change, in order to keep $\mathbf{L}$ conserved. This will show up as the velocity having a component that's not in the xy plane. In other words, she will be now spinning in a non-vertical axis.
As a concluding note, one can't understand this as conservation of momentum 'independently for each axis'. A rotation around 'multiple axes' is really just a rotation around one combined axis, this is the statement of Euler's rotation theorem.
A: The thing that you're actually seeing in the video is the angular velocity, not the angular momentum. Unlike linear momentum and velocity which are related by a constant scalar, the mass ($\mathbf p=m\mathbf v$), the angular velocity and angular momentum are related by the moment of inertia:
$$\mathbf L = I \boldsymbol\omega.$$
The moment of inertia differs from the mass in that:

*

*it is a matrix not a scalar, so that the angular velocity and angular momentum need not point in the same direction, and

*it depends on the shape of the body, which means it can change if the diver moves his limbs around.

Together these two facts means that the angular velocity need not be conserved even though angular momentum is.
(An addendum to the second point: even if the object doesn't change shape, if it's not symmetric its moment of inertia still changes as it rotates because the moment of inertia depends on the spatial distribution of mass, and that distribution is changing as the object moves! Another wonderful counterintuitive rotation motion is seen in this video of a rotating piece of pipe.)
A: 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.
A: An human body can be compared to a tennis racket with respect to the $3$ principal axis of inertia.
The smallest one is the line from our feet to the head, because the average distance from the axis to the masses around is the smallest. The biggest one is a axis from back to front passing by the COM (somewhere in the belly I guess). And there is the intermediate one: an axis from side to side also passing by the COM.
It is well known that a racket has a stable rotation around the smaller and bigger axis of inertia. But it is easy to see how easily it flips the initial rotation during the fly, if around the intermediate axis.
The athlets use this property to help in flipping the initial rotation around the intermediate axis.
Of course, we are not rigid bodies, and can also change our configuration on the fly. The effect of bending, so that our length reduces almost by half, not only increases the angular velocity, but also changes the stabiliy relation between the axis.
The angular momentum is a vector, so its $x$, $y$ and $z$ components  doesn't change. But that components are not the momentarily axis of rotation. The later is only constant in the particular case of a rotation around a stable axis of inertia.
A: It’s not possible to change angular momentum unless there exists an external torque since torque $${\large\bf\tau}=\frac{d{\bf L}}{dt}$$
where ${\bf L}$ is angular momentum. Angular momentum is conserved unless there exists an external (unbalanced) torque. So it is also true that the total angular momentum stays the same from the moment they leave the platform till they hit the water, which you also correctly point out.
But also note that angular momentum $${\bf L}=I{\bf\omega}\tag1$$ where $I$ is called the moment of inertia and $\bf\omega$ is the divers' angular velocity, or how fast they are rotating and in what direction they are rotating.
As you have stated, when divers jump off the platform they must apply an initial torque so that they give themselves an initial angular momentum. While it is true that there no longer can be an external torque to change their angular momentum after they've left the platform, by equation (1), the diver must change their moment of inertia to achieve a change in their angular velocity (not in their angular momentum though - the product $I{\bf\omega}$ stays the same even though the magnitude of both $I$ and $\omega$ change).
Since the moment of inertia depends on how mass is located and distributed throughout their body, to change it, the diver must change the position of their legs/arms/body parts, relative to the axis of rotation, which causes a change to the speed and how their bodies rotate.
This is how divers are able to change how they are rotating through the air.
