Has anybody available a proof that the covariance between two variables has always the same sign as Spearman's Rho, assuming that both are not zero, or an explanation / counterexample to show why this is not the case?
I am talking about the "population" (theoretical) magnitudes, not their sample counterparts. Namely, for $X, Y$ two random variables with distribution functions $F_X, F_Y$, and with all needed moments, co-moments, etc, existing,
$$\text{Cov}(X,Y) = E(XY) - E(X)E(Y)$$ while
$$\rho_s(X,Y) = \text{Cov}[F_X(X),F_Y(Y)]$$
I know that if $X,Y$ are Quadrant Dependent ($QD$), positively or negatively, this indeed holds,
$$(X,Y) = QD \implies \text{sign}\left\{\text{Cov}(X,Y)\right\} = \text{sign}\left\{\rho_s(X,Y)\right\}$$
...again, if both are not zero. But what if $QD$ cannot be established or doesn't hold?
What I am eventually after is a proof that if $h(Y)$ is an increasing monotonic transformation of $Y$, then $\text{sign}\left\{\text{Cov}(X,Y)\right\} = \text{sign}\left\{\text{Cov}(X,h(Y))\right\}$. I know that this appears strongly intuitive and even "self-evident", but I could not find such a proof anywhere, neither did I manage to prove it myself. More precisely, what I want to show is that, if both are not zero, they cannot have opposite signs.
Now, since Spearman's Rho is invariant to monotonic transformations we do have $\rho_s(X,Y) = \rho_s(X,h(Y))$, so a way to prove the "same sign" result for the covariances, would be to prove that the covariance has always the same sign as Spearman's Rho, hence this question.
I have found an old beautiful expression for the covariance due to W. Hoeffding that brings the $\text{Cov}$ and $\rho_s$ definitions "very close", but I could not prove the general statement without assuming Quadrant Dependence.
Of course, if someone has something directly on the "same sign" (desired) result for the covariances, it would be equally helpful.
UPDATE
I found this question that is related but not identical. As already mentioned, it modifies my question as follows: "Assume that both measures are not zero. Can they have opposite signs?"