Speaker Height / Tilt

I understand that tweeters sound / measure differently off axis, but I don’t understand how they radiate differently above vs below the axis. 

Also, changing tilt changes the tweeter axis, right?

So, here is the evidence tweeters are in fact directional. From the spec sheet of a typical tweeter:
https://www.madisoundspeakerstore.com/soft-dome-tweeters-vifa/peerless-dx25bg60-04-1-textile-dome-tw...
Notice the frequency response chart. Each line represents a different location off axis.


Next, please refer back to the measurements I pointed from Stereophile.

What happens is that the relative difference in distance to your ears causes constructive and destructive interference.  Speaker designers (like me) have to take the acoustic distance into account when designing crossovers.

D'Appolito is the exception in that above and below the tweeter axis is the same. As I tried to mention above, my comments about listening at or  below the tweeter axis applies only to traditional designs with the tweeter on top. With D'Appolitos it's the same amount of pain, and sometimes more so, below and above.

After he initially introduced his famous configuration, Dr. D'Appolito has suggested fixing this by increasing the steepness of the filters if you use his design. That is, if you'd normally do OK with a 2nd order (12 db/Octave) you should try a 4th order (24 db/Octave).

What erik says " I find a lot of speakers sound better if I listen below the tweeter axis, and sometimes with the tweeters not pointed directly at me"
That is exactly my experience with the Wilson Cubs. They sound best with the tweeters just above my ears & intersecting a couple feet behind my head. Same was true with my Tekton's, except those needed to intersect a few feet behind me (less toe in than the Wilson's).    

So there’s a "right side up" for raw tweeter drivers? If the mounting housing was round or square and symmetrical, you could actually position it 4 different ways and create different frequency responses?? Same true of mid-ranges?

Generally when you sit above the tweeter axis, assuming it has good dispersion you end up with a large hole in the crossover region (2-3K generally). This doesn't exist if you get a coaxial speaker.

Good example is the M106 once you go above axis:  https://www.stereophile.com/images/914Revelfig5.jpg