Question: Assume $X$ and $Y$ are independent random variables. Is $Median(XY) = Median(X) \cdot Median(Y)$? If so, how would one prove this? If not, what conditions would be sufficient for this relationship to hold?
Additional question: Does the relationship hold for $\alpha$-trimmed means?
Update: Based on a conversation with @Glen_b in the comments on his answer, as well as the contribution of @nikie, it appears that sufficient conditions for the relationship to hold are: 1) independence, and 2) at least one of the distributions of $X$ and $Y$ has a median of zero.