Regarding types of gyroballs, you can think about two different variations using the football analogy. Consider the case where a quarterback is throwing a quick slant to his tight end across the middle. He will fire the football straight at his man, snapping his hand down to impart the typical spiral. This is the same basic motion as for a regular gyroball. Now consider the case where the quarterback is throwing a bomb to his wide receiver. While he still throws a spiral, the football leaves his hand at an upward angle, say in the 15-45 degree range. This is comparable to what is referred to as a gyroball with side force. This ball is supposed to drop less than a regular gyroball, but break to the left away from a righthanded hitter. An variation of this is if the pitcher holds the ball longer, such that the spin axis points down instead of up, which is supposed to cause a break to the right. A fourth kind, a gyroball with lift force, is thrown straight toward the plate, but with the hand more behind the ball than on the side, resulting in the spin axis pointing away from the plate toward the 3rd base side. In this case, the ball "rises" as compared to the drop on a regular gyroball. In addition to the different ways to release the ball, these pitches can be thrown in 2-seam or 4-seam versions, which will affect the amount of break/drop. To get an idea of what these pitches do, check out this picture taken from one of the original researcher's videos:

Shown are the end location of four pitches thrown from the same arm slot. The red dot is a fastball, the blue dot is a forkball, the white dot is a regular gyro, and the green dot is a gyro with side force. A gyro with lift force would end up somewhere between the red and white dots.
I wanted to post some links to relevant articles that I have been collecting (for those who are interested).
This paper written by a physics professor at the University of Illinois contains a detailed discussion of what I've covered above.
This site (Google translation from Japanese) has a slide presentation of the findings by the original researchers (Himeno/Tezuka), with
this page having a series of videos presumably part of that presentation.
I also wanted to post this -
a graphic purportedly depicting the throwing motion and break for a gyroball. I believe that this, along with many articles written on the subject, are incorrect. It shows the pitch being thrown with the wrist snapping over the top of the ball (similar to a screwball), imparting a clockwise spin. This disagrees with much of what I have read which indicates that the wrist goes straight down, producing a counterclockwise spin. Also, the graphic says that the pitch will break sharply to the left, which differs from what I described above. Perhaps Kyle can set the record straight here.
Based on these and other articles I have read, here's what I believe the main advantages of throwing the gyro are:
- For a regular gyro, the ball drops similarly to a forkball or split-finger fastball, but at a speed closer to a regular fastball.
- A slight change in grip and wrist action allows the pitcher to reduce the drop or impart sideways motion. This would allow the pitcher to overcome a batter who has learned to recognize the gyro from the different pitching motion, as the different variations probably look almost the same.
- The claim by the researchers that proper double-spin throwing mechanics results in less stress on a pitcher's body.