  We present a polarization resolved study of the low energy band structure in
the optimally doped iron pnictide superconductor
Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$ (T$_c$=37K) using angle resolved photoemission
spectroscopy. Polarization-contrasted measurements are used to identify and
trace all three low energy hole-like bands predicted by local density
approximation (LDA) calculations. The photoemitted electrons reveal an
inconsistency with LDA-predicted symmetries along the $\Gamma$-X high symmetry
momentum axis, due to unexpectedly strong rotational anisotropy in electron
kinetics. We evaluate many-body effects such as Mott-Hubbard interactions that
are likely to underlie the anomaly, and discuss how the observed deviations
from LDA band structure affect the energetics of iron pnictide Cooper pairing
in the hole doped regime.
