Purpose of Dome on Tweeter...

The purpose of the dome shape is rigidity. Domes are very strong for their weight.

At frequencies where the dome is effectively rigid, the dispersion is essentially that of a piston of the same diameter. The dome shape does not improve dispersion in and of itself because it's not a pulsating dome, it's a moving-back-and-forth dome.

At frequencies where the dome is in breakup mode, the dispersion tends towards that of a ring radiator, with the diameter of the ring being the diameter of the voice coil. Some of the inverted dome tweeters have a voice coil that is smaller in diameter than the perimeter of the dome, and therefore have a little bit better dispersion when they go into breakup mode.

A pushed-in tweeter dome no longer has the geometric rigidity it was intended to have, and even if you manage to pop it back into shape, the material may have been weaked where it was creased and so its behavior may be degraded.

Duke
dealer/manufacturer/occasional user of dome tweeters

Duke,

Would there be an advantage to domed woofers?

"Would there be an advantage to domed woofers"?

If it gave the behavior the designer wanted, then yes.

My guess is that there would be a tradeoff, where the dome diaphragm is probably stiffer and remains pistonic to a higher frequency, but then has nastier behavior when breakup finally does set in, compared to a more traditional curvilinear cone. Few choices in loudspeaker design are without tradeoffs.

Morel and Dynaudio make woofers whose voice coils are quite large in diameter relative to their cone diameter, so they end up being quasi-dome woofers. Both companies also specialize in tweeters that can be crossed over unusually low, and this may be an adaptation to the characteristics of quasi-dome woofers made from paper or polymers.

I think Accuton uses a concave dome profile for their ceramic-diaphragm woofers, and this would be consistent with their philosophy of maintaining pistonic behavior as high up as possible. I think Cabasse used to make shallow-dome-profile honeycomb-sandwich woofer cones, but I don't know what material they used.

Duke

I do believe EPI had the first inverted dome tweeter that was not attached to a horn

As Duke mentioned above, Dynaudio & Morel mid/bass drivers with 3 inch voice coils basically have a 3 inch dome type dust cap. I myself am a big fan of the 6 inch Morel doped paper cone. Great controlled bass from the 3 inch VC and that raised dome acts like a 3 inch dome midrange. Check out the ATC monitors, they also have raised dome dust caps.

Fritz, builder of FritzSpeakers