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, see <A HREF="http://static.stevereads.com/papers_to_read/wave_propagation_in_even_and_odd_dimensional_spaces.pdf">Wave Propagation in Even and Odd Dimensional Spaces</A>. 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. (The study of this difference goes back to Volterra and Hadamard.)

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.