Use an [icosahedral sphere](http://gamedev.stackexchange.com/questions/16585/how-do-you-programmatically-generate-a-sphere/60107#60107) instead of a polar-coordinates one.

The net for the icosahedron can be laid out nicely, like this:

![enter image description here][1]

In a texture editing program, it's easy to UVMap an icosahedron with good/regular texel shapes.


  [1]: https://i.sstatic.net/rIpnH.png