Obviously it has to pass through itself. The interesting thing is that it is possible to evert it without pinching or creasing.

Yes, that’s why you can’t do it in 3 dimensions (without self-intersection) - if you have an extra dimension to hand you can invert the sphere using the extra dimension to avoid self-intersection I believe.

Thanks, this confused me greatly watching the graphics, I'm no mathematician... OK if we can just push the sphere through itself then yeah I guess you can invert it...