By looking at the mechanism the E2 mechanism occurs in one step. There is no intermediate. In the E1 mechanism you do have an intermediate that is subject to a rearrangement (methyl or hydride shift)
E1 forms a carbocation, allowing for the shift. E2 is a concerted mechanism - due to lack of electron contribution from branching, you'll never see a separate carbocation forming intermediate, and therefore no chance for a shift.