This is probably a bad thing to post on a thread about visualizing inverting a surface, but since I've been making my clothes for over 2 decades, I had to check this out. So I took off my trousers and pinned the cuffs together along the circumference (it just so happens that I'm making a new winter coat, so I had pins handy), matching the seams to each other. I grabbed the cuffs and pulled one leg up through the other. The result is that the pinned cuffs sit right at the crotch seam -- there is no way to completely invert the trousers any more than this, and I don't think it would be possible even if the pants had a higher lycra content, as there is no way to get the cuffs THROUGH the crotch to complete the inversion. However, if I grab the pinned cuffs through the waist band and flick the trousers away from me, they turn right side out with a lot less effort than it takes to turn them inside out.
I was able to do it with my jeans, so I expect it's possible with any trousers. You should note that the inversion is not symmetrical. You are basically pulling one pant leg into the other one. In the end you are left with one pant leg which is inside-out and straight and the other pant leg is inside of the first.
The inversion of the parameters is now noticeable. You started with basically a donut with a very wide hole (formed by the pant legs -- I'm ignoring the hole where your waist goes). After the inversion, the pant legs form a donut which is very narrow (just the width of the inside of the pant leg) and is very tall (the height of the pant leg).
I stared at the animated example for nearly two hours last night. I did get the one leg inside of the other donut, but it didn't /look/ like the animated example, so I disregarded it. Thank you -- "not symmetrical" did not occur to me -- it was bugging me. There's a lot of this same sort of visual thought that goes into designing various garments, but the surface is not a torus.
Also, all of the cash fell out of my pockets.