  We study the accuracy of spectroscopic RVs by comparing spectroscopic and
astrometric RVs for stars of the Hyades open cluster. Rather accurate
astrometric RVs are available for the Hyades' stars, based on Hipparcos and on
the first Gaia data release. We obtained HARPS spectra for a large sample of
Hyades stars, and homogeneously analysed them. After cleaning the sample from
binaries, RV variables, and outliers, 71 stars remained. The distribution of
the observed RV difference (spectroscopic -- astrometric) is skewed and depends
on the star right ascension. This is consistent with the Hyades cluster
rotating at 42.3 +/- 10 m/s/pc. The two Hyades giants in the sample show, as
predicted by gravitational redshift (GR), a spectroscopic RV that is
blue-shifted with respect to the dwarfs, and the empirical GR slope is of 626
+/- 131 m/s, in agreement with the theoretical prediction. The difference
between spectroscopic and astrometric RVs is very close to zero, within the
uncertainties. In particular, the mean difference is of -33 m/s and the median
is of -16 m/s when considering the Gaia-based RVs (corrected for cluster
rotation), with a sigma of 347 m/s, very close to the expected cluster velocity
dispersion. We also determine a new value of the cluster centroid spectroscopic
RV: 39.36 +/- 0.26 km/s. The spectroscopic RV measurements are expected, from
simulations, to depend on stellar rotation, but our data do not confirm these
predictions. We finally discuss other phenomena that can influence the RV
difference, such as cluster expansion, stellar activity, general relativity,
and Galactic potential. Clusters within the reach of current telescopes are
expected to show differences of several hundreds m/s, depending on their
position in the Galaxy.
