Braiding non-orientable surfaces in S^4

Closed braided surfaces in $S^4$ are the two-dimensional analogous of closed braids in $S^3$. They are usefull in studying smooth closed orientable surfaces in $S^4$, since any such a surface is isotopic to a braided one. We show that the non-orientable version of this result does not hold, that is smooth closed non-orientable surfaces cannot be braided. In fact, any reasonable definition of non-orientable braided surfaces leads to very strong restrictions in terms of self-intersection and Euler characteristic.