You're asking about correcting the "pitch" of the maxilla. I explained the terminology of "pitch," "yaw," and "roll" in a recent post (  /community/community/main-forum/elements-of-smile-aesthetics-and-craniofacial-dystrophy/#post-2419 )

In maxillofacial surgery, they simply detach part of the maxilla and reattach it with the symmetry corrected in all three dimensions. 

In practice, non-surgical repositioning of the maxilla isn't as simple as just correcting the pitch, yaw, and roll at the suture lines because the overall shape of the maxilla itself also seems to change. @zegtv brought up this discussion (  /community/community/main-forum/is-the-front-of-the-palate-being-curved-downwards-an-indicator-of-downward-long-growth/ ) in comparison of the following two pictures:

@progress suggested that transversely expanding the palate so that it becomes less vaulted can alter the pitch of the maxilla. 

Posted by: Progress

It can be an indicator of both undeveloped palate and downward maxillary growth. Contrary to what many intuitively think, palatal expansion causes the posterior palate to swing down and forward in relation to the anterior palate and the rest of the skull. As a consequence, the upper dental arch will gain more height (downward), increasing the length of ramus, and leading to a more horizontal appearance of the jaw along with increased postural suppport. The effect can be simulated by placing a pen between your molars and biting on it. You will probably find it easier to keep your spine sraight and your mandible will look better.

Likewise, a narrow smile is not only the result of molars hiding behind the premolars, but the posterior palate being positioned too UP in the skull. This also implies that if one side of your palate is lower than the other, the lower side may often actually be your more developed side, especially if the tip of your nose is curved towards the opposite, higher side.

So you might have already achieved some of the "rotation" you are asking about through the 6mm of palate expansion you have reported. It is interesting to think about how this realignment might sustainably achieve the effects of starecta splints. I agree with @progress that the roof of the mouth should drop down with expansion, but I'm still not sure that this has any effect on the occlusion/Curve of Spee.

I've only expanded my palate by about 3mm and there's no readily discernible change in the pitch of my maxilla, but there might be some improvement in the roll (i.e. occlusal cant). From the beginning, my gingiva has been visible above my molars and bicuspids when smiling, but not above my canines or incisors, so I think the posterior part of my maxilla might already be lower than the anterior part.

I think you may be right. I see a lot less gum when I smile wide. Used to have to tense my upper lip and artificially lengthen my philtrum when I smiled due to gum exposure. By this I mean I would "tuck" my upper lip in which then would lengthen the philtrum, thus covering the gum. This would also make the upper lip essentially disappear and it looked quite unnatural (to me at least but I doubt anyone noticed)

However, now I find I can let the lip relax and less gum is exposed. It makes sense that flattening the roof of the mouth over time would cause it to "rotate" as the widening essentially pushes the edges of the gum out and up. And I also think the pressure from the tongue is pushing against gravity which may cause the maxilla to resposition itself to its proper place over time