My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a  "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the bottom or collar of the shirt. He thought it was really cool (kids are easily amused, and so am I).

So he learned that you can turn a shirt or pants right-side-out by pulling the material through any hole, not just certain ones. I told him that even if there was a rip in the shirt, you could use that to turn it inside-out or right-side-out, and he was fascinated by this and asked "why?"

I don't really know the answer to this. Why is this the case? What if the sleeves of a long-sleeve shirt were sewn together at the cuff, creating a continuous tube from one sleeve to the other? Would you still be able to turn it right-side-out? Why? What properties must a garment have so that it can be turned inside-out and right-side-out?

Sorry if this is a lame question, but I've always wondered. I wouldn't even know what to google for, so that is why I am asking here.

If you know the answer to this, could you please put it into layman's terms?

Update: Wow, I really appreciate all the participation. This is a really pleasant community and I have learned a lot here. It seems that the answer is that you need at least one puncture in the garment through which to push or pull the fabric. It appears that you can have certain handles, although it's not usually practical with clothing due to necessary stretching.

Accepted (a while ago actually -- sorry for not updating sooner) Dan's answer because among the answers that I understand, it is the highest ranked by this community.

Everything I'm wearing is a topological sphere, with holes (t-shirts have 4, pants 3, shoes and socks 1) in which cases any hole works.

Instead of a long-sleeve shirt with the arms sewn together, consider a pair of pants with legs sewn together to form a topological torus with a hole (so if you were to wear them your feet would be touching and it would be impossible to put your shoes on).  This pair of pants has two parameters which are roughly constant, the circumference of the leg and the total length of the two legs.

When it is turned inside out through the waist, these parameters swap roles, so you will have a tube about the length of a pant leg with an opening on each end, the same as if you had turned one leg inside-out and pushed the other leg through it prior to sewing.

I think this would be possible to do with real toroidal clothing (e.g. a skirt) as long as it is thin enough, because the process doesn't require any stretching.
