Let me try to focus the question as follows: Suppose you wake up in a dark and empty space. Which properties can help you decide whether the space is even or odd-dimensional?

In the context of wave propagation, there is the fundamental difference that <A HREF="https://en.wikipedia.org/wiki/Huygens–Fresnel_principle">Huygens principle</A> is only valid in odd-dimensional space. Sharly defined wavefronts (think of ripples in a pond when you throw a stone in) need an odd number of spatial dimensions, in even-dimensional space the wavefront decays with a long tail.

So to test for even/odd dimensionality you only need to shout in empty space. If you hear an echo you live in an even number of spatial dimensions.