A wireless sensor can detect the presence of an intruder in its sensing range, and is said to cover the portion of a given barrier that intersects with its sensing range. Barrier coverage is achieved by a set of sensors if every point on the barrier is covered by some sensor in the set. Assuming n identical, anonymous, and relocatable sensors are placed initially at arbitrary positions on a line segment barrier, we are interested in the following question: under what circumstances can they independently make decisions and movements in order to reach final positions whereby they collectively cover the barrier? We assume each sensor repeatedly executes Look-Compute-Move cycles: it looks to find the positions of sensors in its visibility range, it computes its next position, and then moves to the calculated position. We consider only oblivious or memoryless sensors with sensing range r and visibility range 2r and assume that sensors can move at most distance r along the barrier in a move. Under these assumptions, it was shown recently that if the sensors are fully synchronized, then there exists an algorithm for barrier coverage even if sensors are unoriented, that is, they do not distinguish between left and right . In this paper, we prove that orientation is critical to being able to solve the problem if we relax the assumption of tight synchronization. We show that if sensors are unoriented, then barrier coverage is unsolvable even in the semi-synchronous setting. In contrast, if sensors agree on a global orientation, then we give an algorithm for barrier coverage, even in the completely asynchronous setting. Finally, we extend the result of  and show that convergence to barrier coverage by unoriented sensors in the semi-synchronous model is possible with bounded visibility range 2r+ρ (for arbitrarily small ρ > 0) and bounded mobility range r.