In the good old days, most geared tuners for guitars and mandolins had the worm gear below the cog, as in the photo here. These are what many refer to as "reverse"tuners.

At some point someone finally realized that gears functioned a little better if the worm was above the cog. Why? Gears always have a bit of backlash. With the cog above the worm, that slop was pulled tight by the string tension. With the cog below the worm, the string tension did not absorb the slop or backlash as well, and staying in tune was more difficult. So it is said—I think it works about the same either way.
 

Sometime around 1925, everyone in the industry agreed to change to the other configuration, and gears started to be made that way. For guitars and simple mandolins, it was no big deal: just move the post holes so the buttons lined up along the sides of the headstock correctly. But on Gibson's F-style mandolins, it really messed with the aesthetics of their headstock. The gear shafts with the worm attached had to stick out just so, and moving the gear posts down the face of the headstock looked rather odd. So being slaves to fashion, they didn't change them for a very long time. And even after they did, they changed back anyway. 

As you can see here, the top treble-side replacement tuner buttons hit the curlicue. If you move the whole gear plate down, the posts lie very oddly on the headstock as seen from the front, which means the posts closest to the nut have to be moved way down and the plate is sometimes projecting off the back of the headstock. The proportions of the headstock are seriously tweaked.

The bass-side tuner posts in the left image are photoshopped to where they'd be with the modern tuners. The treble side is unchanged. The image on right shows a headstock with modern ("worm over") gears. 

Gotoh makes their F-gears both ways. Stew-Mac offers only one of them, the early style:

Gotoh makes them with the worm above and below the cog in a variety of styles. From eBay and elsewhere:

From Roger Siminoff's webpage:

If the plates are aligned, the posts and button shaft positions are therefore different. 

The post location is important, and so is the shaft location. 
 
 
 
 
 

If the posts are in the same location, the plates and button shafts both come out different, as in the photo of the F-5, above. This is also not workable unless the headstock is fundamentally redesigned to accommodate it..
 
 
 

In any case...no matter which type of tuners you have, the buttons should turn the same direction to make the gears turn the right way, like this:

Turn that D button counterclockwise, and it should make the post turn precisely the same direction whether the worm is above or below the cog. Once again: