Graphene is composed of two coupled sublattices and therefore supports an additional quantum number called pseudospin. In bilayer graphene, pseudospin is determined by the relative amplitude of the wave function on the two layers, with "up" pseudospin corresponding to the electronic density on one layer and "down" to the density on the other. We propose a non-magnetic, pseudospin-based version of a spin valve, in which the pseudospin polarization in neighbouring regions of a bilayer is controlled by external gates. Numerical calculations demonstrate a large on-off ratio of such a device. This finding holds promise for the realization of "pseudospintronics": a form of electronics based upon the manipulation of pseudospin analogous to the control of physical spin in spintronics applications.