  We present results from model fitting to the Spectral Energy Distribution
(SED) of a homogeneous sample of Seyfert II galaxies drawn from the $12\mu$m
Galaxy Sample. Imaging and nuclear flux measurements are presented in an
accompanying paper (Videla et al., 2013). Here we add IRS Spitzer observations
to further constrain the SEDs after careful subtraction of a starburst
component. We use the library of CLUMPY torus models from Nenkova et
al.~(2008ab) and also test the two-phase models recently produced by Stalevski
et al.~(2012). We find that photometric and spectroscopic observations in the
mid-IR (>5mu) are crucial to properly constrain the best-fit torus models.
About half of our sources show clear near-IR excess of their SEDs above the
best fit models. This problem can be less severe when using the Stalevski et
al.~(2012) models. It is not clear what is the nature of this emission since
best fitted black body temperatures are very high (~1700-2500 K) and the Type
II classification of our sources would correspond to a small probability to
peer directly into the hottest regions of the torus. Crucially, the derived
torus parameters when using CLUMPY models are quite robust,, independently of
whether the sources require an additional black body component or not. Our
findings suggest that tori are characterized by N_0 > 5, sigma > 40, tau < 25,
i > 40 degrees, Y < 50 and A_v^los ~ 100-300. From these we can determine that
typical torus sizes and masses of 0.1-5.0 pc and 10^{4-6} M_\odot. We find
tentative evidence that those nuclei with a detected Hidden Broad Line Regions
are characterized by lower levels of extinction than those without one.
Finally, we find no correlation between the torus properties and the presence
of circumnuclear or more global star-formation.
