This paper looks at asymmetry in the even and odd multipoles of the CMB. These correspond to the symmetric and antisymmetric part of the temperature anisotropy [tex]T^+(\hat{n})=\frac{1}{2}[ T(\hat{n}) + T(\hat{n}) ][/tex] and [tex]T^(\hat{n})=\frac{1}{2}[ T(\hat{n})  T(\hat{n}) ][/tex], respectively, where [tex]\hat{n}[/tex] is a unit vector.
The authors find that there is more power on odd multipoles than on even multipoles, for the range [tex]2\leq\ell\leq22[/tex]. Comparing the WMAP data with simulations of the [tex]\Lambda[/tex]CDM model, the authors find that the probability of finding as strong asymmetry as present in the data is 0.6\% for WMAP3, 0.4\% for WMAP5 and 0.3\% for WMAP7.
As possible reasons, the authors discuss asymmetric beam shape, known foregrounds, the skycut, and other effects. They conclude that none of these can explain the data. In terms of the primordial power spectrum, the asymmetry would correspond to perturbations where the real part is strongly suppressed compared to the imaginary part (so that the symmetric cosine part is amplified with respect to the asymmetric sine part).
An important issue is the a posteriori choice of the multipole range. The authors address this by doing simulations and retaining only the ones for which the probability is lowest at [tex]\ell_{max}=22[/tex]. They find that still only 2\% of the simulations have as much asymmetry as the data. I don't think I understand how this entirely addresses the issue, though.
In a followup paper, 1011.0377, the authors connect the evenodd asymmetry to the difference between predicted and observed angular correlation function. It is well known that the observed correlation function is basically zero for angles [tex]\theta[/tex] between 60 and 150 degrees, and [tex]\theta>150^\circ[/tex], while the [tex]\Lambda[/tex]CDM prediction is negative for [tex]60^\circ<\theta<120^\circ[/tex] and positive for [tex]\theta>120^\circ[/tex]. Asymmetry in the above multipole range happens to map neatly to the behaviour of the correlation function, so they are a mnifestation of the same thing  whether it is a statistical fluke or something else.
[1002.0148] Anomalous parity asymmetry of WMAP power spectrum data at low multpoles: is it cosmological or systematics?
Authors:  Jaiseung Kim, Pavel Naselsky 
Abstract:  We have investigated the oddparity preference of the WMAP 7 year power spectrum. Comparison with simulation shows the oddparity preference of WMAP data (2<= l <=22) is anomalous at 4in1000 level. We have investigated its origins, and ruled out some of noncosmological origins such as asymmetric beams, noise and cutsky effect. We also find primordial origin requires Re[\Phi(\mathbf k)]\llIm[\Phi(\mathbf k)] for k\lesssim 22/\eta_0, where \eta_0 is the present conformal time. Multipoles associated with the oddparity preference happen to coincide with some of other CMB anomalies. Therefore, there may exist a common origin, whether cosmological or not. Besides, we find it likely that low quadrupole power is the part of this oddparity preference anomaly rather than an isolated one. The Planck surveyor, which possesses wide frequency coverage and systematics distinct from the WMAP, may allow us to resolve the mystery of the anomalous oddparity preference. 
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[1002.0148] Anomalous parity asymmetry of WMAP power spectr
It is always hard to assess the effect of a posteriori choice (the multipole range l_{max} in our case).
Later, we realized a posteriori choice on l_{max} is rather unimportant, because the asymmetry exists over wide range of l_{max}.
To be specific, according to Fig. 4 in the followup paper (1006.1979) by a different group, the statistical significance is relatively unaffected as far as
l_{max} lies between 15 and 27.
For instance, choosing l_{max}=22 barely enhance the statistical significance, compared to that of l_{max}=20.
Therefore, 98\% significance mentioned in the additional analysis of our work (1002.0148), with a posterior choice taken into account, was somewhat underestimated.
Later, we realized a posteriori choice on l_{max} is rather unimportant, because the asymmetry exists over wide range of l_{max}.
To be specific, according to Fig. 4 in the followup paper (1006.1979) by a different group, the statistical significance is relatively unaffected as far as
l_{max} lies between 15 and 27.
For instance, choosing l_{max}=22 barely enhance the statistical significance, compared to that of l_{max}=20.
Therefore, 98\% significance mentioned in the additional analysis of our work (1002.0148), with a posterior choice taken into account, was somewhat underestimated.