As a consequence of the open mapping theorem, a continuous linear bijection H:X->Y between Banach spaces X and Y must be a linear homeomorphism. The main result of this article (Theorem 9) is similar in form but makes no continuity assumptions on H: If X and Y have symmetric Schauder bases (see before Theorem 9 for the definition), then a ''basis separating'' linear bijection H is a linear homeomorphism. Given Banach spaces X and Y with Schauder bases {x n } and {y n }, respectively, we say that H:X->Y, H( n N x(n)x n )= n N Hx(n)y n , is basis separating if for all elements x= n N x(n)x n and y= n N y(n)x n of X, x(n)y(n)=0 for all n N implies that Hx(n)Hy(n)=0 for all n N. We show that associated with a linear basis separating map H, there is a support map h:N->N ~ . The support map enables us to develop a canonical form (Theorem 4) for basis separating maps that plays a crucial role in the development of the main results.