If we have that $X$ and $Y$ are independent, what do we know about there components $X^\pm, Y^\pm$. Are they independent as well? If so, which combinations are the independent ones and which ones are not and why?
For obvious reasons $X$ and $Y$ take on values on the entire real line.
Any combination yields an independent pair. If $X$ and $Y$ are independent random variables and $f$ and $g$ are measurable functions, then $f(X)$ and $g(Y)$, too, are independent. And $f(x)=x\vee0$, for example, is measurable. Intuitively, if there's nothing to be learned about $Y$ from $X$, then nor is there anything to be learned about $Y^+$ from $X^+$, etc.