The other day I was explaining orientability to someone and we were walking through some of the statements about orientability on the Wikipedia page on the topic. While I was able to satisfy his curiosity, one statement on that page (which I did not even attempt to delve into with him) has been nagging me since then:

"For example, a torus in $K^2\times S^1$ can be one-sided and a Klein bottle in the same space can be two-sided."

Because this statement bothered me (since it runs counter to normal intuition about orientable surfaces in Euclidean spaces), I have been thinking about it more over the last few days. I have been able to determine which copies of these submanifolds should have the stated properties and convince myself how the non-orientability of the ambient space $K^2\times S^1$ allows for the submanifolds in question to twist back on themselves in unusual ways, but nevertheless I still cannot form a decent picture of what this really means.

The real issue with my understanding what is going on with these submanifolds seems to be that although these phenomenon occur in a non-orientable space, this space can itself be embedded in an orientable space and so it seems that these odd tori and Klein bottles should therefore embed in an orientable space as well and so I should have some chance of visualizing these phenomena when I project down to $\mathbb{R}^2$ or $\mathbb{R}^3$

Question:Does anyone have a good picture or other approach to help visualize what a one-sided torus or two-sided Klein bottle looks like?

So while it may be too much to hope for a projection that accurately reflects the sidedness of these creatures, I am hoping someone may have a decent projection of either of these creatures to the plane or 3-space that shows some manifestations of their odd behaviour in their ambient space. Or, barring an actual picture, perhaps someone who has thought about this more has some other way of thinking about them which at least gives a better intuitive sense of how to look at them in their ambient space and 'see' (whatever that may mean when you think about them) these counterintuitive features.

normalintuition about orientable surfaces" - he he. Puns. $\endgroup$ – Reinstate Monica Sep 21 '17 at 2:21