It is well known that odd and even dimensions work differently. 

* [Waves propagation in odd dimensions is unlike propagation in even dimensions.](https://math.stackexchange.com/questions/8794/intuitive-explanation-of-the-difference-between-waves-in-odd-and-even-dimensions)
* [A parity operator](http://en.wikipedia.org/wiki/Parity_%28physics%29#Effect_of_spatial_inversion_on_some_variables_of_classical_physics) is a rotation in even dimensions, but cannot be represented as a rotation in odd dimensions.
* [Even dimensional spheres do not admit continuous nonvanishing vector fields](http://en.wikipedia.org/wiki/Hairy_ball_theorem), but odd dimensional spheres do.

Is there some unifying reason for these differences? Why is dimensional parity so important?

(If you know of any more interesting differences, please leave a comment and I will add them to the list.)