How Do I Turn Around?

Check this picture out to see how it can be done

Think of a cat turning around inside of a bent tube, while the tube itself remains stationary in free space. To get a better idea, check this GIF of a bent-axis hydraulic motor. As the fluid pressures cause parts of it to turn internally, the bent exterior frame of the motor does not move, i.e., from the outside it appears to be stationary. What makes it possible for the the cat, the astronaut, and the hydraulic motor to rotate in free space is the fact they are all bent.

Competition springboard divers use this technique in order to do twists (long axis of rotation through body), which requires at least a brief moment of being bent during the dive, in order to "snap a twist".

Edit:

Consider two identical cylinders of mass $m$ each, radius $r$ and height $h$ . An abstract massless bent axle passes through both symmetrically as in the graphic, with an axis passing through the center of mass of each, making an angle of $\theta$ with the bent axis on each side.

It is assumed that work can be done either between each cylinder and the massless bent axle or between the cylinders themselves to produce a rotation of the cylinders about the abstract bent axle--same result either way. But we are restricting this analysis to solid bodies that don't actually have internally an independently spinning real axle with a mass. The abstract massless bent axle is for purposes of analysis only. Note that in the analysis that will follow, even if the axle was real but not bent, and is massless, then any work done between the cylinder(s) and the axle will result in infinite angular speed of the axle while producing no change in the angular speed of the cylinder(s). Hence the need for a bent abstract massless axle. Even a massless axle "acquires a non-zero momentum of inertia" in being bent, because of the existence of the other cylinder of non-zero mass on it, even if the cylinder is freely spinning on it.

The moment of inertia of a cylinder about the cylinder’s axis of rotation is

$\dfrac{1}{2}m{r}^{2}$

while about the axis perpendicular to the cylinder’s axis of rotation through the center of mass is

$\dfrac{1}{12}m \left( 3{r}^{2}+{h}^{2} \right)$

From this, given the angle $\theta$ , the angular momentum of both cylinders about that axis passing through the center of mass of each is (Technical note: This simple vector combination is possible because the cylinder exhibits symmetry about all three $x, y, z$ planes---normally for non-symmetric solids it's more complicated)

$m {\omega}_{1} \left({r}^{2} Cos\left(\theta\right) +\dfrac{1}{6}\left(3{r}^{2}+ {h}^{2}\right)Sin\left(\theta\right) \right)$

while the combined angular moment of both cylinders is (Vector Addition of Angular Momentum)

$m{\omega}_{2} {r}^{2} Cos\left(\theta\right)$

where ${\omega}_{1}$ , ${\omega}_{2}$ are the angular speeds of the assembly about the axis passing through the center of mass of the cylinders, and the angular speed of each cylinder about the bent axis, respectively.

Since total angular momentum is conserved, the ratio of the absolute angular speeds is then

$\dfrac { { \omega }_{ 1 } }{ { \omega }_{ 2 } } =\dfrac { 6{ r }^{ 2 }Cos\left( \theta \right) }{ 6{ r }^{ 2 }Cos\left( \theta \right) +\left( 3{ r }^{ 2 }+{h}^{2} \right) Sin\left( \theta \right) }$

which ranges from $1$ to $0$ as $\theta$ ranges from $0$ to $90$ degrees. For a typical example, for $\theta =30$ degrees and letting the ratio $\dfrac{h}{r}=6$ , the ratio works out to about $0.21$ .

While angular momentum is conserved, angular kinetic energy is not. Work is done in bringing the two cylinders up to angular speed about the bent axle, which will also result in a rotation in the opposite direction about the axis passing through the center of mass of both cylinders, but at a slower angular speed. Work is done in reversing everything to a full stop, but because the differences in the angular speeds ${\omega}_{1}$ , ${\omega}_{2}$ , the orientation of the bent axes and the cylinders may not necessarily be the same as before.

The ability of competition divers, ice skaters and ballerinas to be able to "snap a twist", leaving them spinning in midair, requires an even lengthier explanation, but it starts here.