It is shown how standard treatments of the change in a vector Δξi under parallel transport about a closed path in Riemannian spacetime lead to erroneous results if a complete circuit is made rather than just half a circuit followed by antisymmetrization. In fact a nonzero change in the vector is obtained upon returning to the initial point via the reversed path. The error in these treatments is shown to involve several aspects of the law of transplantation for a vector and the idea of path reversal in curved spaces.