What would it feel as a human being to experience the Dzhanibekov Flip? The Tennis Racket theorem describes the following effect: rotation of an object around its first and third principal axes is stable, while rotation around its second principal axis (or intermediate axis) is not.
This gives rise to an interesting phenomenon called the Dzhanibekov Effect, discovered in 1985 in a space station, in which a t-bar rotating in zero gravity along its intermediate axis flips around magically (without any torque or force applied), cf. e.g. this Phys.SE post.

I've been wondering how would it feel as a human being to experience the "Dzhanibekov flip". Since human beings are asymmetric and have 3 different moments of inertia, we should flip if rotated around the intermediate axis. This experience would, of course, be performed in space (in zero gravity). As I thought about this, I realized that we wouldn't feel any difference at all. Since angular momentum is conserved (it has to be! there is no torque), we would feel we're still rotating the same. And yet, someone else watching us would see us flip.
So, my question is this: Am I thinking correctly above about the human experience of the Dzhanibekov Flip? Because this is somewhat intuition-defying. That t-bar in the animation above is certainly flipping. Would it not be cognizant of its flip if it were a human?
 A: You would definitely feel the flip in your guts, with an intensity depending on how close you get to the intermediate axis and your initial spin. While the principle of relativity says that you cannot feel velocity, you can feel acceleration with respect to an inertial frame, in particular rotation. Intuitively, your guts would be pushed around by inertia, so you should be able to detect whether you are spinning and the direction of the axis of rotation.
I do not quite understand your argument of conservation of angular momentum. While it is true in the observer's inertial frame, this does not mean that in your frame the angular momentum (and angular velocity which you feel) the stays fixed.
Assimilating a human by a brick (or using approximate planar reflections), your intermediate axis would be about waist going left/right. Your angular momentum would start roughly in this direction and the flip would reorient its direction, staying on your front/back side to end up in the opposite orientation. This variation of angular momentum would be accompanied by a noticeable variation in angular velocity.
Hope this helps.
A: About rotation of rigid objects in general.
Let me start with a simpler case: a case that is generally known as 'Feynmans wobbling plate'.
The reason for discussing that case:
In the case of Feynman's wobbling plate there is no external torque, yet the plate (if thrown in a particular way) does not rotate in a simple way.
(It is possible of course, to throw the plate in such a way that it rotates in a simple way. The point is, the plate does not necessarily rotate in a simple way.)
Youtube video with 40 second simulation:
Feynman's wobbling plate, in space frame and body frame
See also this youtube video, with an actual plate actually thrown: Feynman's wobbling plate

The wobbling motion is a process of internal relocation of momentum. Momentum gets transferred from one part of the plate to another, continuously. When the thrown plate is wobbling like that the internal stresses are quite high.
The wobbling motion can continue forever, but only when there is no dissipation of kinetic energy. In the real world objects always flex a little, and that dissipates kinetic energy.
(For an example of dissipation of kinetic energy affecting a pattern of rotation: the story of the satellite explorer I )

The Dzhanibekov effect
The motion pattern in the case of the Dzhanibekov effect is more complicated than the motion pattern in the case of Feynman's wobbling plate because in the case of the Dzhanibekov effect there are three different moments of inertia.
What the Dzhanibekov effect has in common with Feynman's wobbling plate is that angular momentum is internally relocated. So there are significant internal stresses.

So:
Can a human performer demonstrate the Dzhanibekov effect using his own body as the rotating body?
If it can be done at all then only with great effort. (And of course it can only be demonstrated in weightless conditions, hence only onboard a space station.)
The performer would need to spin quite fast, and it very hard to keep your limbs together when spinning that fast.
Additionally, the performer would have to keep his body very rigid, otherwise kinetic energy would dissipate. Dissipation of kinetic energy changes the motion pattern, so it would make the demonstration less convincing.
(The performer can try the following: rotate slow, record the demonstration on video, and play the video faster. But then the performer would have to keep his body rigid for a loooong time.)

See also:
Youtube video with simulation of an object undergoing Dzhanibekov effect motion. The motion is explained in detail.
Rigid body motion and the Dzhanibekov effect
