Gyrotropy in twisted bilayer graphene can be used as a signature of interlayer electronic coupling. Gyrotropy can emerge in the absence of interlayer coupling in time-reversal symmetric bilayer systems. This gyrotropy originates from the extrinsic geometry associated with the physical geometry of the system and is independent of the structure of the electronic states. We first illustrate this effect for a purely classical bilayer array of one-dimensional wires. Next we study pristine twisted bilayer graphene and show that the gyrotropy is entirely due to interlayer coupling. Finally we consider twisted bilayer MoTe2, first as a pristine model where the gyrotropy exactly vanishes, and then with weak strain and displacement fields where we show that the geometric gyrotropy can dominate the gyrotropy due to interlayer coupling. Our results call attention to the necessity to separate the contribution of extrinsic physical geometry from the contribution of intrinsic electronic states to the properties of twisted materials.