Observing the motion of the solar system and galaxy through space? This is very perplexing.
Conditions:

planets are moving around the sun (solar system)
solar system is moving as part of the galaxy
the galaxy is moving through space on some axis
galaxy is part of a cluster, also moving through space
other stars and constellations are also moving through space, presumably in the same direction and speed as the Milky way.

I appreciate the answer to Why is the Solar Helical (Vortex) model wrong? - but considering the question: how can we observe the solar system's Helical motion through space, I have a new question... please allow me to present both, since they are related.
Question 1:
Again, how can we observe the Helical motion of our solar system (and galaxy) from earth? Is it by observing the passing of planets across/through constellations?
Question 2:
If I can make an analogy: like different ticking clocks stored inside a grandfather clock - is the solar system following its own Helical motion and therefore are all other solar systems within the Milkyway also following their own Helical motion? AND is the galaxy also following its own Helical motion!?
OR!
Is the solar system's motion fixed (Helically speaking), and any such movement is tied to the motion of the parent, i.e. the galaxy? In other words, is the Helical motion of the solar system the same as the galaxy, or are all solar systems moving Helically through space, e.g. many clocks within a grandfather clocks analogy.
Initially, the question was simple: how can be observe the solar system's Helical motion?
But it seemed unintuitive to me that the solar system is moving Helically while the galaxy is also moving as such - both structures on their own trajectory and paths - seems like galactic chaos.
And, all the while planets maintain their fixed orbits.
I would have guessed that the solar system is NOT moving Helically, but the galaxy is, and the solar system is simply a fixed orbital passenger, as so to speak - in the same way 8 grains of sand (planets) are passengers in a bowl of jelly (galaxy), both being carried by another vehicle (cluster).
It seems to me then, the solar system's motion through space is limited by the galaxy's motion, in terms of axis, speed - and therefore the galaxy is also fixed like a passenger on a much larger galaxy cluster.
It seems reasonable to suggest the whole universe is moving Helically(?), and all galaxies merely inherit this, but are more like grains of sand in jelly than independently ticking clocks within a larger grandfather clock.
If our solar system is moving Helically independently from the galaxy, the solar system could move at its own speed, either quicker or slower depending on environmental factors. Could the solar system really skip-out of the galaxy, racing ahead on its own Helical path(?).
So many questions! But Question 1 is the one I wish to request help answering.
All motion is relative. There is no absolute space which can be said to be the correct reference frame in which we must measure all motion. Hence, for any object you're free to construct any reference frame that does, or does not, give that object a helical motion.
Reference frames
Of course, some reference frames make more sense than others. When you cycle to work, it makes sense to measure your speed relative to the bike lane. The reference frame of the Sun makes sense if you're concerned with planets' orbit. In this frame, Earth has no helical motion, only circular motion. Or you could call it helical motion with $v_z = 0$, if you wish.
On the other hand, the Sun and its neighbors follow a path through the Milky Way; the so-called Local Standard of Rest, measured by observing nearby stars and star-forming regions. In this frame, Earth has a (non-$v_z\!\!=\!0$) helical motion.
Similarly, in the reference frame of Milky Way's path toward, say, Andromeda, the Sun has a helical motion. Another star will have a similar, but different, motion.
Comoving coordinates
That said, there is one reference frame which has a special status, namely the Universe's comoving frame. These are the coordinates that expand along with the Universe, and they are the coordinates in which everything on average is at rest. If you hear the phrase &quot;Milky Way's motion through space&quot;, this is probably what is referred to. Since time flows differently in different frames, it's also the frame of choice when you talk about the age of the Universe. Likewise, since lengths are contracted in moving frames, the comoving frame is the frame in which we measure cosmological distances.
It's important to realize, however, that motions, positions, etc. measured in the comoving frame are no more &quot;correct&quot; than in any other frame.
The cosmic microwave background
But how can we measure a velocity in this reference frame, if all motion is relative?
One way would be to measure the motion of a million galaxies with respect to us, and take the average. But a much easier method, which is in fact how we do it, is to measure the dipole of the cosmic microwave background (CMB). This radiation was emitted when the Universe was very young, and has the special property that it was emitted isotropically (i.e. the same in all directions) with a near-perfect Planck spectrum.
Hence, if you standing still in the comoving frame, you will see exactly the same CMB spectrum in all directions, whereas if you're moving you will observe the CMB to be blueshifted in front of you, and redshifted behind you. This difference between the two opposite directions is called the dipole, and its value was measured by the Planck satellite (Planck Collaboration 2020) to imply a total velocity of the Sun in the comoving reference frame of
$$
v_\odot = 369.82\pm0.11\,\mathrm{km}\,\mathrm{s}^{-1},
$$
toward the Galactic coordinates
$$
\{\ell,b\} = \{264.021^\circ\pm0.011^\circ, 48.253^\circ\pm0.005^\circ\}.
$$
Note the amazing precision, with an uncertainty in speed and direction of only 400 km/h and less than an arcminute, respectively!
This velocity is important to know, since it is used to correct all measured cosmological velocities.