  We present results from simultaneous modeling of 2D (projected along the line
of sight) position, proper motion and line-of-sight velocity for \textit{Gaia}-
and APOGEE-observed stars near the centre of the Sagittarius (Sgr) dwarf
spheroidal galaxy. We use a mixture model that allows for independent
sub-populations contributed by the Sgr galaxy, its nuclear star cluster M54,
and the Milky Way foreground. We find an offset of $0.295\pm 0.029$ degrees
between the inferred centroids of Sgr and M54, corresponding to a (projected)
physical separation of $0.135\pm 0.013$ kpc. The detected offset might
plausibly be driven by unmodelled asymmetry in Sgr's stellar configuration;
however, standard criteria for model selection favour our symmetric model over
an alternative that allows for bilateral asymmetry. We infer an offset between
the proper motion centres of Sgr and M54 of
$[\Delta\mu_{\alpha}\cos\delta,\Delta\mu_{\delta}]=[4.9, -19.7] \pm [6.8, 6.2]$
$\mu$as yr$^{-1}$ ($[0.61, -2.46] \pm [0.85, 0.77] $ km s$^{-1}$), with
magnitude similar to the covariance expected due to spatially-correlated
systematic error. We infer an offset of $4.1\pm 1.2$ km s$^{-1}$ in
line-of-sight velocity. Using inferred values for the systemic positions and
motions of Sgr and M54 as initial conditions, we calculate the recent orbital
history of a simplified Sgr/M54 system, which we demonstrate to be sensitive to
any line-of-sight distance offset between M54 and Sgr, and to the distribution
of dark matter within Sgr.
