  We investigate the radial velocity (RV) variability and spectroscopic
binarity of 19 Galactic long-period ($P_{\rm{puls}} \gtrsim 10$ d) classical
Cepheid variable stars whose trigonometric parallaxes are being measured using
the Hubble Space Telescope and Gaia. Our primary objective is to constrain
possible parallax error due to undetected orbital motion. Using $>1600$
high-precision RVs measured between 2011 and 2016, we find no indication of
orbital motion on $\lesssim 5$ yr timescales for 18 Cepheids and determine
upper limits on allowed configurations for a range of input orbital periods.
The results constrain the unsigned parallax error due to orbital motion to $< 2
\%$ for 16 stars, and $< 4 \%$ for 18. We improve the orbital solution of the
known binary YZ Carinae and show that the astrometric model must take into
account orbital motion to avoid significant error ($\sim \pm 100 \mu$arcsec).
We further investigate long-timescale ($P_{\rm{orb}} > 10$ yr) variations in
pulsation-averaged velocity $v_\gamma$ via a template fitting approach using
both new and literature RVs. We discover the spectroscopic binarity of XZ Car
and CD Cyg, find first tentative evidence for AQ Car, and reveal KN Cen's
orbital signature. Further (mostly tentative) evidence of time-variable
$v_\gamma$ is found for SS CMa, VY Car, SZ Cyg, and X Pup. We briefly discuss
considerations regarding a vetting process of Galactic Leavitt law calibrators
and show that light contributions by companions are insignificant for most
distance scale applications.
