  We prove that spin chains symmetric under a combination of mirror and
spin-flip symmetries and with a nondegenerate spectrum show finite spin
transport at zero total magnetization and infinite temperature. We demonstrate
this numerically using two prominent examples: the Stark many-body localization
system and the symmetrized many-body localization system. We provide evidence
of delocalization at all energy densities and show that the delocalization
mechanism is robust to breaking the symmetry. We use our results to construct
two localized systems which, when coupled, delocalize each other.
