  We study the planar regime of curvature perturbations for single field
inflationary models in an axially symmetric Bianchi I background. In a theory
with standard scalar field action, the power spectrum for such modes has a pole
as the planarity parameter goes to zero. We show that constraints from back
reaction lead to a strong lower bound on the planarity parameter for
high-momentum planar modes and use this bound to calculate the signal-to-noise
ratio of the anisotropic power spectrum in the CMB, which in turn places an
upper bound on the Hubble scale during inflation allowed in our model. We find
that non-Gaussianities for these planar modes are enhanced for the flattened
triangle and the squeezed triangle configurations, but show that the estimated
values of the f_NL parameters remain well below the experimental bounds from
the CMB for generic planar modes (other, more promising signatures are also
discussed). For a standard action, f_NL from the squeezed configuration turns
out to be larger compared to that from the flattened triangle configuration in
the planar regime. However, in a theory with higher derivative operators,
non-Gaussianities from the flattened triangle can become larger than the
squeezed configuration in a certain limit of the planarity parameter.
