  The complex interplay between turbulence, magnetic fields, and self-gravity
leads to the formation of molecular clouds out of the diffuse interstellar
medium (ISM). One avenue of studying this interplay is by analyzing statistical
features derived from observations, where the interpretation of these features
is greatly facilitated by comparisons with numerical simulations. Here we focus
on the statistical anisotropy present in synthetic maps of velocity centroid
data, which we derive from three-dimensional magnetohydrodynamic simulations of
a turbulent, magnetized, self-gravitating patch of ISM. We study how the
orientation and magnitude of the velocity anisotropy correlate with the
magnetic field and with the structures generated by gravitational collapse.
  Motivated by recent observational constraints, our simulations focus on the
supersonic (sonic Mach number $\mathcal{M} \approx 2 - 17$) but sub- to
trans-alfvenic (alfvenic Mach number $\mathcal{M}_A \approx 0.2 - 1.2$)
turbulence regime, and we consider clouds which are barely to mildly
magnetically supercritical (mass-to-flux ratio equal to once or twice the
critical value). Additionally we explore the impact of the turbulence driving
mechanism (solenoidal or compressive) on the velocity anisotropy.
  While we confirm previous findings that the velocity anisotropy generally
aligns well with the plane-of-sky magnetic field, our inclusion of the effects
of self-gravity reveals that in regions of higher column density, the velocity
anisotropy may be destroyed or even reoriented to align with the
gravitationally formed structures. We provide evidence that this effect is not
necessarily due to the increase of $\mathcal{M}_A$ inside the high-density
regions.
