Why does symmetry lead to a restoring force in precessional rotation?

[sirchasm]

I know it's just a mental block - for some reason I can't get the phrase "restoring force" out of the ongoing debate in my noggin. But this is a question about symmetry specifically, and the notion of a force appearing because of an asymmetry, or "symmetry-breaking".

The simplest notion I have of the principle of broken symmetry is a coupled pendulum. A single string pendulum with one point of connection is more symmetrical than a pendulum hanging from a spring coupled at either end (which is another pendulum).
With the constrained motion introduced to the "system", the pendulum will "relax' into a mode that's a linear, perpendicular one - to the line made by the spring length. The spring breaks the symmetry, and "absorbs" any motion of the pendulum except transverse to the plane formed by it + the hanging weight. Any motion of the weight parallel to the plane of the spring does work on the spring, making it "rotate' in the plane, and energy is lost to frictional coupling etc. This is the "resulting force" from the asymmetry.

How does the symmetry argument apply to precessional rotation? A gyroscope that isn't spinning is more symmetrical in the inertial field of planet earth. The symmetry is broken when it's given some angular momentum - but the precession rate is independent of the rotation of the inertial field - i.e. planet earth?
I'm looking for a succinct explanation of the symmetry (of precessional rotation) and what breaks it - without referring to the torque as a time derivative except as a result of the asymmetry produced by the angular momentum of the precessing body...?

 

[sirchasm]

OK - another stab.

The geometrical view of a free string pendulum is a cone, or a conic section that traces out a cone, or an elliptic cone.
With a constraining spring, the conic symmetry is "forced" to be a 2-d section of the available space.
A free pendulum precesses as the global inertial frame rotates, or it rotates along 2 axes independently of the global frame. A constrained pendulum precesses because of the spring, and rotates depending on the spring and its 2-d frame (the spring alters the local geometry of the inertial frame). Independent rotational momentum is "broken".

The geometry of a gyroscope is usually a rigid disk, which rotates along 3 axes, so the space is spherical. Precession looks like a torus or a toroid. How is a constraint introduced or a local geometry altered by giving the wheel some spin?

 

[sirchasm]

I get the idea there's a problem here - a gyroscope with zero angular momentum isn't a gyroscope or a pendulum.

If the symmetry is inertial, what local symmetry is altered to explain the appearance of torque? The torque a free swinging pendulum sees can be explained by a global rotating frame - as the weight moves, the global frame shifts by some slight angle, so the weight stops or inverts its motion at a point that isn't diametrically opposite its last point of inversion, or in the global frame it moves in slightly curved rather than straight paths.

The torque experienced by the axis of a rotating disk is usually explained as the time derivative of the sum of inertial moments of the rotating mass itself, not as a local asymmetry in a global inertial frame which is also rotating - but the rotations are independent yet "coupled" (a string pendulum is coupled, a gyroscope has a stand or a spinning bike wheel has someone holding it) to an independently rotating inertial frame, the planet.

 

[sirchasm]

To connect the precession of a free pendulum to the idea of gyroscopic precession, simply have a gyroscope as a free pendulum weight.

Then there's two precessional torques, one because it's a free-swinging weight, and one because it has gyroscopic motion; so how is a gyroscopically precessing body that's also a free pendulum (has a single thread or string of some kind attached to a body that it rotates in respect of) a broken symmetry, or is it not broken?
If the spinning weight isn't gimballed but the axis is directly coupled, there's a torque along the string, which should twist, or standing waves start up, maybe.