Klein tunneling through an asymmetric barrier: Symmetric transmission and directional pair creation

We prove that the transmission probability for the Klein tunneling through a spatially asymmetric barrier is the same for left and right incidence whenever each asymptotic lead carries a single propagating channel per direction. Time-dependent Wigner-function simulations confirm this and locate the missing directionality in the barrier's interior, where a sharp edge generates several times more under-barrier negative-energy population than a smooth one. Directional control in the Klein regime therefore resides in pair production rather than in the transmitted current.