Based on my finding this additional definition for axisymmetrical, "exhibiting cylindrical symmetry", only a cylindrical/spherical/hemispherical pattern would fit the definition.
Yep, and my added 2c...
The cylindrical symmetry part is essential to the axisymmetric definition.
I think what makes axisymmetric confusing, is we tend to think of two axes extending from the speaker, horizontal and vertical.
But neither of these axes are the ones that apply to whether a speaker is axisymmetric.
The axis that matters, is the axis extending straight from the speaker to the listener or measurement mic.
Where there will be 360 degree pattern symmetry around that axis.
So in the cases of an MTM, or a synergy with a non-square horn, or a coax with non-square or non-round horn, etc, etc,
none of them would be axisymmetric.
