That is the effect I already mentioned. The attitude of the bullet is the slight upwards tilt of the nose compared with the trajectory at that instant. As gravity starts to win and the bullet drops, so the attitude diverts more and more from the trajectory. The ability of the projectile to alter its attitude to follow the trajectory is termed tractability. But the bullet will always have some attitude angle, because it is always dropping under the influence of gravity. This means that the lower surface has a higher air pressure on it than the top surface. Putting it totally non-scientifically, the bullets "bites " into the air more on the lower surface, and thus tends to crawl in the direction of the spin.
A cricket ball has a very rough surface, and the seam makes a ring of extreme roughness that is used by a spin bowler to achieve a similar effect (generating different air pressure on different sides) and generate a curved ball. That's why they do all that curious polishing of one side on their trousers. Must be similar for a baseball pitcher. And if anyone ignorant of cricket or baseball finds that all a bit far-fetched, then watching over-the shoulder TV pictures of Shane Warne in action some years ago was a real eye-opener for me and anyone else with an interest in ballistics who was fortunate enough to see it. The bamboozling effect on the batsman was (wicket-)shattering.
If you can read German, then I recommend the volumes on ballistics by Beat P.Kneubuehl. There must be something similar in English. Not light reading, but it gets your ballistics knowhow off the anecdotal level pretty steeply.
Sorry if that was a bit involved, but you did ask!