On a characterization question for symmetric random variables

If X₁, X₂ are independent with common density g symmetric about zero, then $\text{P(X}{}_1\text{+ αX}{}_2\text{> 0) =}\text{1}\text{2}$ for all real α. We provide a counter example to show that the converse is false and thus settle a question posed by Burdick (1972).