It is well known that under mild assumptions a scalar diffusion $dX_t = a(X_t) dt + \sigma(X_t) dW_t$ with invariant probability distribution $\pi$ is reversible. This is indeed not true for multidimensional diffusions. The usual proofs consists in writing down generators, speed functions etc...

I am trying to intuitively understand this result, and the only (not very satisfying) argument that I have found is the following. It is straightforward to check that any Markov chain on $\mathbb{Z}$ that has an invariant probability $\pi$ and that can only make jumps of size $+1$ or $-1$ is reversible: notice for example that $$F(k) = \pi(k)p(k,k+1)-\pi(k+1)p(k+1,k)$$ is independent of $k$ and is thus equal to $0$. If $a(\cdot)$ and $\sigma(\cdot)$ are regular enough, a diffusion can be seen as a limit of such Markov chains on $\epsilon \mathbb{Z}$ so that this makes the result plausible.

**question**: what are arguments/proofs/examples that could shed light on why a one dimensional ergodic diffusion is automatically reversible.