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 Huygens principle is only valid in odd-dimensional space, see Wave Propagation in Even and Odd Dimensional Spaces. Sharply 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.