The quest for molecular structures exhibiting strong quantum interference effects in the transport setting has long been on the forefront of chemical research. Here, we establish theoretically that the unusual geometry of spiro-conjugated systems gives rise to complete destructive interference in the resonant-transport regime. This results in a current blockade of the type not present in meta-connected benzene or similar molecular structures. We further show that these systems can undergo a transport-driven Jahn-Teller distortion which can lift the aforementioned destructive-interference effects. The overall transport characteristics is determined by the interplay between the two phenomena. Spiro-conjugated systems may therefore serve as a novel platform for investigations of quantum interference and vibronic effects in the charge transport setting. The potential to control quantum interference in these systems can also turn them into attractive components in designing functional molecular circuits.