Graphene electrons feature a pair of massless Dirac cones of opposite pseudospin chirality at two valleys. Klein tunneling refers to the intriguing capability of these chiral electrons to penetrate through high and wide potential barrier. The two valleys have been treated independently in the literature, where time reversal symmetry dictates that neither the normal incidence transmission nor the angle-averaged one can have any valley polarization. Here we show that, when intervalley scattering by barrier is accounted, graphene electrons normally incident at a superlattice barrier can experience a fully valley-selective Klein tunneling, i.e. perfect transmission in one valley, and perfect reflection in the other. Intervalley backscattering creates staggered pseudospin gaps in the superlattice barrier, which, combined with the valley contrast in pseudospin chirality, determines the valley polarity of Klein tunneling. The angle averaged transmission can have a net valley polarization of 20% for a 5-period barrier, and exceed 75% for a 20-period barrier. Our finding points to an unexpected opportunity to realize valley functionalities in graphene electronics.
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