I found an interesting video on turning a sphere inside out.

https://www.youtube.com/watch?v=wO61D9x6lNY

I was thinking about whether using these rules it would be possible to turn hyperspheres inside out and I think I found out one general method for turning an odd dimensional hypersphere inside out but I'm still not sure whether or not it's possible to turn an even dimensional hypersphere with more than two dimensions inside out using the rules described in the video.

In any number of dimensions except one a hypersphere can have (n-1)/2 perpendicular circles on it that don't intersect and the hypersphere will have two poles that are parallel to all the perpendicular circles on it that don't intersect. In order to turn an odd dimensional hypersphere inside out divide it into guide segments connected by wavy segments and have the segments extending from pole to pole and extend the segments through all the circles parallel to the polls that are on the hypersphere. Push the poles through each other and then rotate the two poles along every independent direction of rotation that exist in its number of dimensions and rotate each of the two poles in opposite directions in every dimension of rotation.