The future space-based gravitational wave detector laser interferometer space antenna (LISA) requires bidirectional exchange of light between its two optical benches on board of each of its three satellites. The current baseline foresees a polarization-maintaining single-mode fiber for this backlink connection.Phase changes which are common in both directions do not enter the science measurement, but differential ('non-reciprocal') phase fluctuations directly do and must thus be guaranteed to be small enough.We have built a setup consisting of a Zerodur baseplate with fused silica components attached to it using hydroxide-catalysis bonding and demonstrated the reciprocity of a polarization-maintaining single-mode fiber at the 1 pm $\sqrt{{\rm Hz}}^{-1}$ level as is required for LISA. We used balanced detection to reduce the influence of parasitic optical beams on the reciprocity measurement and a fiber length stabilization to avoid nonlinear effects in our phase measurement system (phase meter). For LISA, a different phase meter is planned to be used that does not show this nonlinearity. We corrected the influence of beam angle changes and temperature changes on the reciprocity measurement in post-processing.