Why can you turn clothing right-side-out?

First, a warning. I suspect this response is likely not going to be immediately comprehensible. There is a formal set-up for your question, there are tools available to understand what's going on. They're not particularly light tools, but they exist and they're worthy of being mentioned. Before I write down the main theorem, let me set-up some terminology. The tools belong to a subject called manifold theory and algebraic topology. The names of the tools I'm going to use are called things like: the isotopy extension theorem, fibre-bundles, fibrations and homotopy-groups.

You have a surface $\Sigma$, it's your shirt or whatever else you're interested in, some surface in 3-dimensional space. Surfaces have automorphism groups, let me call it $\operatorname{Aut}(\Sigma)$. These are, say, all the self-homeomorphisms or diffeomorphisms of the surface. And surfaces can sit in space. A way of putting a surface in space is called an embedding. Let's call all the embeddings of the surface $\operatorname{Emb}(\Sigma, \mathbb R^3)$. $\operatorname{Emb}(\Sigma, \mathbb R^3)$ is a set, but in the subject of topology these sets have a natural topology as well. We think of them as a space where "nearby" embeddings are almost the same, except for maybe a little wiggle here or there. The topology on the set of embeddings is called the compact-open topology (see Wikipedia, for details on most of these definitions).

Okay, so now there's some formal nonsense. Look at the quotient space $\operatorname{Emb}(\Sigma, \mathbb R^3)/\operatorname{Aut}(\Sigma)$. You can think of this as all ways $\Sigma$ can sit in space, but without any labelling -- the surface has no parametrization. So it's the space of all subspaces of $\mathbb R^3$ that just happen to be homeomorphic to your surface.

Richard Palais has a really nice theorem that puts this all into a pleasant context. The preamble is we need to think of everything as living in the world of smooth manifolds -- smooth embeddings, $\operatorname{Aut}(\Sigma)$ is the diffeomorphism group of the surface, etc.

There are two locally-trivial fibre bundles (or something more easy to prove -- Serre fibrations), this is the "global" isotopy-extension theorem:

$$\operatorname{Diff}(\mathbb R^3, \Sigma) \to \operatorname{Diff}(\mathbb R^3) \to \operatorname{Emb}(\Sigma, \mathbb R^3)/\operatorname{Aut}(\Sigma)$$

$$\operatorname{Diff}(\mathbb R^3 \operatorname{fix} \Sigma) \to \operatorname{Diff}(\mathbb R^3, \Sigma) \to \operatorname{Aut}(\Sigma)$$ here $\operatorname{Diff}(\mathbb R^3)$ indicates diffeomorphisms of $\mathbb R^3$ that are the identity outside of a sufficiently large ball, say.

So the Palais theorem, together with the homotopy long exact sequence of a fibration, is giving you a language that allows you to translate between automorphisms of your surface, and motions of the surface in space.

It's a theorem of Jean Cerf's that $\operatorname{Diff}(\mathbb R^3)$ is connected. A little diagram chase says that an automorphism of a surface can be realized by a motion of that surface in 3-space if and only if that automorphism of the surface extends to an automorphism of 3-space. For closed surfaces, the Jordan-Brouwer separation theorem gives you an obstruction to turning your surface inside-out. But for non-closed surfaces you're out of tools.

To figure out if you can realize an automorphism as a motion, you literally have to try to extend it "by hands". This is a very general phenomena -- you have one manifold sitting in another, but rarely does an automorphism of the submanifold extend to the ambient manifold. You see this phenomena happening in various other branches of mathematics as well -- an automorphism of a subgroup does not always extend to the ambient group, etc.

So you try your luck and try to build the extension yourself. In some vague sense that's a formal analogy between the visceral mystery of turning the surface inside-out and a kind of formalized mathematical problem, but of a fundamentally analogous feel.

We're looking for automorphisms that reverse orientation. For an arbitrary surface with boundary in 3-space, it's not clear if you can turn the surface inside out. This is because the surface might be knotted. Unknotted surfaces are examples like your t-shirt. Let's try to cook up something that can't be turned inside-out.

The automorphism group of a 3-times punctured sphere has 12 path-components (12 elements up to isotopy). There are 6 elements that preserve orientation, and 6 that reverse. In particular the orientation-reversing automorphisms reverse the orientation of all the boundary circles. So if you could come up with a knotted pair-of-pants (3-times punctured surface) so that its boundary circles did not admit a symmetry that reversed the orientations of all three circles simultaneously, you'd be done.

Maybe this doesn't seem like a reduction to you, but it is.

For example, there are things called non-invertible knots:

http://en.wikipedia.org/wiki/Invertible_knot

So how do we cook up a knotted pair-of-pants from that?

Here's the idea. The non-invertible knot in the link above is sometimes called $8_{17}$. Here is another picture of it:

http://katlas.org/wiki/8_17

Here is a variant on that.

Interpret this image as a ribbon of paper that has three boundary circles. One boundary circle is unknotted. One is $8_{17}$. The other is some other knot.

It turns out that other knot isn't trivial, nor is it $8_{17}$.

So why can't this knotted pair of pants be turned inside-out? Well, the three knots are distinct, and $8_{17}$ can't be reversed.

The reason why I know the other knot isn't $8_{17}$? It's a hyperbolic knot and it has a different ($4.40083...$) hyperbolic volume than $8_{17}$ ($10.9859...$).

FYI: in some sense this is one of the simplest surfaces with non-trivial boundary that can't be turned inside-out. All discs can be turned inside-out. Similarly, all annuli (regardless of how they're knotted) can be turned inside-out. So for genus zero surfaces, 3 boundary components is the least you can have if you're looking for a surface that can't be turned inside-out.

edited to correct for Jason's comment.

comment added later: I suggest if you purchase a garment of this form you return it to the manufacturer.

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.

I'm going to try to give a lighter-flavoured version of my previous answer. I'd rather not edit the previous one anymore so here goes another response. I want to make clear, this response is to you, not your 10-year-old nephew. How you translate this response to any person depends more on you and that person than anything else.

Take a look at the Wikipedia page for diffeomorphism. In particular,the lead image

When I look at that image I see the standard Cartesian coordinate grid, but deformed a little.

There's a "big theorem" in a subject called Manifold Theory and it's name is the "Isotopy Extension Theorem". Moreover, it has a lot to do with these kinds of pictures.

The isotopy extension theorem is roughly this construction: say you have some rubber, and it's sitting in a medium of liquid epoxy that's near-set. Moreover, imagine the epoxy to be multi-coloured. So when you move the rubber bit around in the epoxy, the epoxy will "track" the rubber object. If your epoxy had a happy-face coloured into it originally, after you move the rubber, you'll see a deformed happy-face.

So you get images that look a lot like mixed paint. Stir various blotches of paint, and the paint gets distorted. The more you stir, the more it mixes and it gets harder and harder to see the original image. The important thing is that the mixed paint is something of a "record" of how you moved your rubber object. And if your motion of the rubber object returns it to its initial position, there is a function

$$ f : X \to X $$

where $X$ is all positions outside your rubber object. Given $x \in X$ you can ask where the particle of paint at position $x$ went after the mixing, and call that position $f(x)$.

All my talk about fibre bundles and homotopy-groups in the previous response was a "high level" encoding of the above idea. An intermediate step in the formalization of this idea is the solution of an ordinary differential equation, and that differential equation is essentially the "paint-mixing idea" above, in case you want to look at this subject in more detail later.

So what does this mean? A motion of an object from an initial position back to the initial position gives you an idea of how to "mix paint" outside the object. Or said another way, it gives you an Automorphism of the complement, in our case that's a 1-1, continuous bijective function between 3-dimensional space without the garment and itself.

You may find it odd but mathematicians have been studying "paint mixing" in all kinds of mathematical objects, including "the space outside of garments" and far more bizarre objects for well over 100 years. This is the subject of dynamical systems. "Garment complements" are a very special case, as these are subsets of 3-dimensional euclidean space and so they're 3-manifolds. Over the past 40 years our understanding of 3-manifolds has changed and seriously altered our understanding of things. To give you a sense for what this understanding is, let's start with the basics. 3-manifolds are things that on small scales look just like "standard" 3-dimensional Euclidean space. So 3-manifolds are an instance of "the flat earth problem". Think about the idea that maybe the earth is like a flat sheet of paper that goes on forever. Some people (apparently) believed this at some point. And superficially, as an idea, it's got some things going for it. The evidence that the earth isn't flat requires some build-up.

Anyhow, so 3-manifolds are the next step. Maybe all space isn't flat in some sense. That's a tricky concept to make sense of as space isn't "in" anything -- basically by definition whatever space is in we'd call space, no? Strangely, it's not this simple. A guy named Gauss discovered that there is a way to make sense of space being non-flat without space sitting in something larger. Meaning curvature is a relative thing, not something judged by some exterior absolute standard. This idea was a revelation and spawned the idea of an abstract manifold. To summarize the notion, here is a little thought experiment.

Imagine a rocket with a rope attached to its tail, the other end of the rope fixed to the earth. The rocket takes off and goes straight away from the earth. Years later, the rocket returns from some other direction, and we grab both loose ends of the rope and pull. We pull and pull, and soon the rope is tight. And the rope doesn't move, it's taut. as if it was stuck to something. But the rope isn't touching anything except your hands. Of course you can't see all the rope at one time as the rope is tracing out the (very long) path of the rocket. But if you climb along the rope, after years you can verify: it's finite in length, it's not touching anything except where it's pinned-down on the earth. And it can't be pulled in.

This is what a topologist might call a hole in the universe. We have abstract conceptions of these types of objects ("holes in the universe") but by their nature they're not terribly easy to visualize -- not impossible either, but it takes practice and some training.

In the 1970's by the work of many mathematicians we started to achieve an understanding of what we expected 3-manifolds to be like. In particular we had procedures to construct them all, and a rough idea of how many varieties of them there should be. The conjectural description of them was called the geometrization conjecture. It was a revelation in its day, since it implied that many of our traditional notions of geometry from studying surfaces in 3-dimensional space translate to the description of all 3-dimensional manifolds. The geometriztion conjecture was recently proven in 2002.

The upshot of this theory is that in some sense 3-dimensional manifolds "crystalize" and break apart in certain standard ways. This forces any kind of dynamics on a 3-manifold (like "paint mixing outside of a garment") to respect that crystalization.

So how do I find a garment you can't turn inside-out? I manufacture one so that its exterior crystalizes in a way I understand. In particular I find a complement that won't allow for this kind of turning inside-out. The fact that these things exist is rather delicate and takes work to see. So it's not particularly easy to explain the proof. But that's the essential idea.

Edit: To say a tad more, there is a certain way in which this "crystalization" can be extremely beautiful. One of the simplest types of crystalizations happens when you're dealing with a finite-volume hyperbolic manifold. This happens more often than you might imagine -- and it's the key idea working in the example in my previous response. The decomposition in this case is very special as there's something called the "Epstein-Penner decomposition" which gives a canonical way to cut the complement into convex polytopes. Things like tetrahedra, octahedra, icosahedra, etc, very standard objects. So understanding the dynamics of "garments" frequently gets turned into (ie the problem "reduces to") the understanding of the geometry of convex polytopes -- the kind of things Euclid was very comfortable with. In particular there's software called "SnapPea" which allows for rather easy computations of these things.

_{(source: utk.edu)}

Images taken from Morwen Thistlethwaite's webpage. These are images of the closely-related notion of a "Dirichlet domain".

Here is an image of the Dirichlet domain for the complement of $8_{17}$, the key idea in the construction of my previous post.

Dirichlet domain for the complement of $8_{17}$

Technically, this in the Poincare model for hyperbolic space, which gives it the jagged/curvy appearance.