Radio telescopes with dual linearly polarized feeds regularly participate in Very Long Baseline Interferometry (VLBI). One example is the VLBI Global Observing System (VGOS), which is employed for high-precision geodesy and astrometry. In order to achieve the maximum signal-to-noise ratio, the visibilities of all four polarization products are combined to Stokes I before fringe-fitting. Our aim is to improve cross-polarization bandpass calibration, which is an essential processing step in this context. Here we investigate the shapes of these station-specific quantities as a function of frequency and time. We observed the extra-galactic source 4C 39.25 for six hours with a VGOS network. We correlated the data with the DiFX software and analyzed the visibilities with PolConvert to determine the complex cross-bandpasses with high accuracy. Their frequency-dependent shape is to first order characterized by a group delay between the two orthogonal polarizations, in the order of several hundred picoseconds. We find that this group delay shows systematic variability in the range of a few picoseconds, but can remain stable within this range for several years, as evident from earlier sessions. On top of the linear phase-frequency relationship there are systematic deviations of several tens of degrees, which in addition are subject to smooth temporal evolution. The antenna cross-bandpasses are variable on time scales of ∼ 1 hour, which defines the frequency of necessary calibrator scans. The source 4C 39.25 is confirmed as an excellent cross-bandpass calibrator. Dedicated surveys are highly encouraged to search for more calibrators of similar quality.