Are there examples of non-orientable manifolds in nature?

The real projective plane is the space of orientations for "nematic liquid crystals": these are materials (often found in your TV or computer screen!) composed of molecules shaped roughly like rods, which can point in any direction in 3D. However, they have no head or tail, so two antipodal orientations are identified. We can model nematic liquid crystals thus by a map from $U\subset \mathbb{R}^3$ to $\mathbb{RP}^2$.

The topology of the real projective plane thus comes into play when one thinks about "topological defects" in these materials. A topological defect is a sort of singularity, where in some tubular neighborhood of this defect the material is continuous, but at the points of the defect, there is a discontinuity. Furthermore, this defect is topological, in that it cannot be homotoped away locally.

With a bit of oversimplifying, $\pi_1(\mathbb{RP}^2)=\mathbb{Z}_2$ means that there is one nontrivial type of line defect (since $S^1$ surrounds a line) and $\pi_2(\mathbb{RP}^2)=\mathbb{Z}$ means that there are an infinite number of types of point defects in 3 dimensional nematic liquid crystals.

Here's a schematic image of a cross section of a line defect and a corresponding path on $\mathbb{RP}^2$ corresponding to a circuit around it. These are both from Jim Sethna's article "Order Parameters, Broken Symmetry, and Topology":

Here's an old photograph of droplets of nematic liquid crystal between crossed polarizers from the paper P. Poulin, H. Stark, T. C. Lubensky, and D. A. Weitz, Novel Colloidal Interactions in Anisotropic Fluids. J. Science (1997) vol. 275 page 1770.. I won't say too much about the colors, but they correspond roughly to the orientation of the molecule. The sharp points at the center of each droplet are one or more point defects, discontinuities in orientation. The dark brush-like structures coming out of each point are the regions where molecules are oriented in directions parallel to either of the polarizers - thus it's kind of like the inverse image of two different points on $\mathbb{RP}^2$.

Roughly speaking, a homotopy class of a map from a 1- or 2-sphere to the projective plane being nontrivial, means that the defect cannot be smoothed away (otherwise there would be a homotopy to a constant).

This is part of a much bigger picture of course; and there are other nonorientable spaces that describe the order of materials. I've been vague above because all of this is explained quite beautifully in the article by N.D. Mermin, The topological theory of defects in ordered media Rev. Mod. Phys. 51, 591–648 (1979). For a quicker introduction, the paper cited above "Order Parameters, Broken Symmetry, and Topology" by Jim Sethna (published in 1991 Lectures in Complex Systems, Eds. L. Nagel and D. Stein, Santa Fe Institute Studies in the Sciences of Complexity, Proc. Vol. XV, Addison-Wesley, 1992) covers the basics.

I love this stuff, so let me know if you have any questions and we can correspond further.

There was a study where they took thousands of digital pictures of "natural images", rendered them in grayscale, and looked at all the 3x3 pixel squares which arose in such pictures. Using topological data analysis they found that (after some normalizations) their data points actually clustered around a Klein bottle embedded in the 7 sphere! Here's a paper that talks about it, and tells you where to look for the Klein bottle:

http://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf

It seems that nature "is" a Klein bottle in the following sense. There is a growing field in applied topology (yes, I said that) which goes by "topological data analysis" or sometimes "persistent homology". As I understand it, it works like this: take a manifold embedded in Euclidean space and suppose you have a way to extract a huge number of sample points from the manifold. With enough sample points it is in principle possible to build a simplicial approximation to the manifold which is refined enough to capture its local structure, and one ought to be able to recover actual topological invariants of the manifold by computing them for these approximations and letting the number of sample points approach infinity. This might be how the human eye assembles intuition for the geometry of an object from just looking at it, a process which collects a very large but finite number of sample points.

Now here's a crazy idea. Let's crank things up a notch and imagine the following experiment: go out in the world, take a huge number of high resolution photographs of images of clouds and trees and other natural objects, break them down into patches of 3 pixels by 3 pixels, and treat each patch as a point in $\mathbb{R}^9$. In other words, we are sampling the subspace of $\mathbb{R}^9$ of "natural images". Question: what would the topological structure of this space be? Crazy, right?

Well, this experiment has actually been done and it turns out that the space of natural images is our good old non-orientable friend, the Klein Bottle:

Who would've thought?

References: Informal, Formal