Mapping wave forms to sphere, does wave form y=0 have a reflection?

[bahamagreen]

TL;DR Summary

Continuous mapping of all continuous wave forms to the surface of a sphere

Zero does not have an inverse.
And y=0 does not have an inverse.
Does the wave form y=0 for all x have an inverse?

 

[BvU]

What conditions should an inverse of a 'wave form' fulfil ?


$\$

 

[bahamagreen]

What conditions should an inverse of a 'wave form' fulfil ?


$\$

My first thought is inverse polarity, but if the inverse wave form comprises -y values, any y=0 wouldn't have an inverse -y, so maybe the inverse wave form would not be strictly continuous? That seems asymmetric with respect to inversion...

edit-- I'm using the wrong term, inversion swaps the range and domain, what I am asking about is reflection about the x axis... y -> -y
This is in the context of antipodean pairs on the sphere

 

[bahamagreen]

More checking, but it looks like zero does have reflection...