  Mild, unavoidable deviations from circular-symmetry of instrumental beams in
current Cosmic Microwave Background (CMB) experiments now pose a significant
challenge to deriving high precision inferences from the high sensitivity and
resolution of CMB measurements. We present analytic results, verified by
numerical simulations, that CMB maps of cosmological signal that respect
underlying statistical isotropy (SI) symmetry, measured with an instrument that
has mildly non-circular (NC) beam would, nevertheless, exhibit SI violation.
Further, we show that appropriate observable measures constructed within the
Bipolar spherical harmonic (BipoSH) representation of SI violation capture
subtle NC-beam effects coupled with the scan strategy of the instrument.
  Accompanying their latest 7-year data release, the WMAP team published very
high significance measurements of non-zero BipoSH spectra, A^{20}_{l l} and
A^{20}_{l-2 l}, in the "W" and "V" band of the experiment. We present a strong
case that the BipoSH measurement are primarily explained by the quadrupolar
(m=2) component of the NC-beams, b_{l2}, of the respective channels. The fact
that subtle levels of non-circularity, e.g., WMAP beams |b_{l2}|/b_{l0} < 0.01,
lead to measurable BipoSH spectra points to the immense promise and potential
of the BipoSH representation. The key result of this work is that using BipoSH
measurements it is possible to estimate an equivalent single hit,'parallel
transported' effective NC-beam that matches both the angular power spectrum and
the non-zero BipoSH measurements of the observed maps. Hence, BipoSH analysis
provides a very simple and effective characterization of CMB maps made with
NC-beam. (Abridged)
