TITLE: What is the 'non-intuitive' part in sphere eversion (turning inside out)?
QUESTION [9 upvotes]: The question does not mean sphere eversion is intuitive to me! In fact, it is just the opposite and that is the purpose of this question. 
Recently, I was reading about Smale's paradox, the problem of sphere eversion (turning a sphere inside out). The wiki article is quite clear and gave me a good overview of the topic. I happened to see an animation of the eversion process as well. 
The problem of sphere eversion is to construct a homotopy between the inside and outside of a sphere in a three dimensional space. During the continuous deformation self-intersections of the sphere are allowed and creating creases is not allowed. 
Given that we can self-intersect the sphere while the process of eversion what could be a possible obstruction to the eversion? What exactly do we mean by self-intersection? Moreover, I find it difficult to imagine why a similar process cannot be employed in the circle case? Why can't we self intersect a circle with itself to turn it inside out? Is there an easy explanation for this phenomenon? 
This topic is new to me. I hope the question is not too naive.  Thank you in advance.

REPLY [15 votes]: Watch Outside In (something we should all do anyway, to commemorate Bill Thurston's passing).
To understand the mathematics behind sphere eversions, you should first get a good intuition for the concepts of immersion and regular homotopy. I recommend Guillemin and Pollack's "Differential Topology" book for starters.
To see why it is not possible to turn a circle inside out, you should read up on the Whitney-Graustein Theorem. It basically boils down to the calculation $\pi_1(S^1)=\mathbb{Z}$, once you notice that the normalized differential of an immersion $S^1\looparrowright \mathbb{R}^2$ is a map $S^1\to S^1$.
You'll find a few more resources related to sphere eversions on my web page.