I'd be inclined to say no, Im looking for the answer myself. But
if you have Cov(A-B,A+B)=Cov(A,A)-Cov(B,B)-Cov(B,A)+Cov(A,B), then
the last two will cancel but if Var(B)>Var(A) then we would get
a negative covariance. [Cov(A,A)=Var(A)] So it looks possible
because as far as I know there is no squaring of the coefficeients
when you bring them out of the covariance so a negative answer is
entirely possible.