Simultaneous VLBA polarimetric observations of the v=$\{$1,2$\}$ J=1-0
  and v=1, J=2-1 SiO maser emission toward VY CMa II: component-level
  polarization analysis
  This paper presents a component-level comparison of the polarized v=1 J =1-0,
v=2 J=1-0 and v=1 J=2-1 SiO maser emission towards the supergiant star VY CMa
at milliarcsecond-scale, as observed using the VLBA at $\lambda=7$mm and
$\lambda=3$mm. An earlier paper considered overall maser morphology and
constraints on SiO maser excitation and pumping derived from these data. The
goal of the current paper is to use the measured polarization properties of
individual co-spatial components detected across multiple transitions to
provide constraints on several competing theories for the transport of
polarized maser emission. This approach minimizes the significant effects of
spatial blending. We present several diagnostic tests designed to distinguish
key features of competing theoretical models for maser polarization. The number
of coincident features is limited by sensitivity however, particularly in the
v=1 J=2-1 transition at 86 GHz, and deeper observations are needed. Preliminary
conclusions based on the current data provide some support for: i)
spin-independent solutions for linear polarization; ii) the influence of
geometry on the distribution of fractional linear polarization with intensity;
and, iii) $\pi/2$ rotations in linear polarization position angle arising from
transitions across the Van Vleck angle ($\sin^2{\theta}=2/3$) between the maser
line-of-sight and magnetic field. There is weaker evidence for several
enumerated non-Zeeman explanations for circular polarization. The expected 2:1
ratio in circular polarization between J=1-0 and J=2-1 predicted by standard
Zeeman theory cannot unfortunately be tested conclusively due to insufficient
coincident components.
