Integrating functional molecules into single-molecule devices is a key step toward the realization of future computing machines based on the smallest possible components. In this context, photoswitching molecules that can make a transition between high and low conductivity in response to light are attractive candidates. Here we present the synthesis and conductance properties of a new type of robust molecular photothermal switch based on the norbornadiene (NB)-quadricyclane (QC) system. The transport through the molecule in the ON state is dominated by a pathway through the π-conjugated system, which is no longer available when the system is switched to the OFF state. Interestingly, in the OFF state we find that the same pathway contributes only 12% to the transport properties. We attribute this observation to the strained tetrahedral geometry of the QC. These results challenge the prevailing assumption that current will simply flow through the shortest through-bond path in a molecule.