Deflection Effects on a Spinning Solar Sail Suppose I have a solar-sail-powered starship flying directly away from a star. The sail is flat and perpendicular to the direction of travel. Now, in order to make the trajectory more stable (we think), we make the ship spin around its center of mass and the axis aligned with motion.


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*Would the sail's spin cause a transfer of linear/angular momentum between the sail itself and the impacting radiation, actually causing the trajectory to deviate from the original line?  


My guess in this case (perpendicularly hitting radiation) would be NO, as any effect on one part of the sail would be countered by those on the opposite part (if radiation is uniformly distributed).


*But then, what would happen if the ship was flying across a star system with its sail set at an angle, i.e. only partially facing the star (in this case the side of the sail closer to the star and moving, say, up would receive more radiation than the opposite one, moving down and farther away)? Would there be linear momentum exchange here, causing the trajectory to veer (like a spinning ball moving through the air)?

*What would happen in the case of pure electromagnetic radiation vs. the case where we have massive particles in the star wind?
 A: The answers to all three of your questions are "there is no effect due to the rotation."
A mirror moving parallel to its own surface does not cause any Doppler shift in the reflected light, or any Doppler aberration in the angle of reflection.  This is an exercise in Zangwill's Modern Electrodynamics (Problem 22.21, to be precise), and can be proven (rather tediously) by taking an arbitrary incident wave four-momentum $p_I^\mu$, boosting it into the frame of the mirror, reflecting the appropriate component to get the reflected wave four-vector in the mirror system, and then "unboosting" back into the lab frame to find the four-momentum of the reflected photon $p_R^\mu$.  
This result implies that photons hitting any patch of a rotating mirror will impart exactly as much (three-)momentum as they would if they were hitting a stationary mirror, and so the light pressure exerted on each patch of the mirror is exactly the same.  There will be a slight change in the pressure when the sail is partly facing away from the star, but that will be due to the sail not "catching" as much photon flux due to its angle, not due to the rotation of the sail. 
This result does not rely on the particles being massless, or even on them colliding elastically with the sail (as, again, you can prove rather tediously using the above technique.)  
A: This is a loose answer, not aimed at getting the bounty. Just creating a clearer mental image. 
A spinning light sail is something like a spinning hovering frizbee. If you knock it upward in the center, it goes up. 
Consider a flat sail traveling directly away from a star, and something shades the left half, but not the right half. That would be something like a spinning hovering frizbee, and you knock one edge upward. That side goes up and it wobbles.
Consider a frizbee, and you sprinkle sticky sand uniformly on it. Sand has mass, so the frizbee slows down to conserve angular momentum. 
Light carries momentum without carrying mass. So I expect that it would not change the rotation rate of the sail for that reason. However, light also carries angular momentum. So circularly polarized light (more photons with one spin orientation) would affect angular momentum. Please comment if this is incorrect.
A: In condition one it would not impart a side vector, but it would impart a side vector in condition two. 
Think of it as the direction the craft is "pushed" is halfway between where the light is coming from and where it ends up.
If the light comes from the sun, 0', and is reflected directly back toward the sun, 360', the difference is 180' and the craft is pushed directly away from the sun. If the sail is angled 45' and the sunlight can be seen from the tangent of the craft's orbit or 270' then the vector of the spacecraft will be 135'. I think the photon formulas are actually more complex than I am making them, but the general concept stands.
Condition three is a completely different question and non-photonic energy is usually not included in solar sail calculations. In most solar sails, if an atom size or larger particle hit it, it would damage the sail, reducing overall performance even if it gave a minutely larger impulse for a moment.
