Instead of a rotating blade, simplify the mental picture to a 1x1 meter square sail. The sail is operating in a region of space with a photon density of 1 photon / m3. The photons are all moving in the same direction, perpendicular to the sail, so the sail is hit by 3x108 photons / second. Each photon has momentum p and reflects away with momentum -p so the momentum transfer to the sail is 6x108p / second.
Now replace the photons in this mental picture with stationary particles, and imagine the sail is moving through the particles at 3x108 m/s. As each particle hits the sail, it bounces away at 6x108 m/s which is equivalent to a momentum of 2p. The momentum transfer to the sail in this scenario is also 6x108p. I realize this second mental picture is non-physical as it neglects relativistic effects, but it's easy to see, in a Newtonian sense, that these two scenarios are equivalent. Only the frame of reference has changed.
Now let's modify the second scenario by angling the sail, and moving it sideways at the same time we move it forward. At the same time we're moving the sail forward by 3x108 meters, we'll also move it sideways by 108 meters. The angle of the sail will be set so that the particles still impact it at a right angle, but now that it's moving at a diagonal, it's going to be moving at 3.16x108 m/s, so the particles will be bouncing off at 6.32x108 with momentum 2.11p. Also, the sail will now be sweeping 3.16x108 m3 and so impacting the same number of particles, so the total momtum transfer to the sail will be 6.66x108p. Of course this momentum transfer is now angled instead of straight back so we'd have to split the vector into perpendicular components and find that the rearward component is 6.32x108
Now change back to the original reference frame and the scenario looks like a sail with a 3:1 angle moving sideways at 1/3 c while being hit with photons with momentum p and the forward momentum transfer to the sail is 6.32x108p
Now back up to the original helio-gyro. 1x1 meter sail I've been describing could be the last meter of a very long blade. Long enough that the tangential velocity of the blade tip can be treated as straight sideways movement.