We investigate the electronic structure of graphene stacks having an ordered sequence of pairs of twisted layers - the graphene twist stack. We find that this remarkable system entails a fundamental mixing of dimensionalities: While the twist stack spectrum is generated by an ensemble of independent effective twist bilayer Hamiltonians, the wave functions are products of bilayer wave functions and standing electron waves in the stacking direction, and thus extend over many layers of the stack. These have the property that of the ensemble of Dirac cones that constitute the twist stack band structure, it is those topologically closest to single layer graphene that dominate the surface region. We further examine the impact of stacking disorder, finding that these results are robust for moderate stacking fault density.