Cointegration is not "directional" because its defining property is intrinsically "nondirectional": a linear combination of the original, integrated series must be a stationary series (here I disregard cointegration of higher orders for simplicity). There is nothing directional in this definition.
However, if the linear combination (more precisely, the coefficients on the different variables) is not known in advance, it has to be estimated. One way of doing that is by regressing one variable on the others by ordinary least squares. This particular estimation technique yields an effect of "direction"; it appears from the formulation of the estimation problem. The errors of $y$ projected on $x$ are not the same as those of $x$ projected on $y$. This is an unfortunate property of the estimation method (unfortunate in the context of cointegration analysis but not necessarily beyond). The problem should go away asymptotically, I think, but it can cause some trouble in finite samples, just as you have found out.
Alternatively, a single cointegrating vector can be estimated as the last principal component (the one with the smallest variance) of the system of variables. Principal component analysis is a "nondirectional" technique so there is no such problem there.