We study the power asymmetry between even and odd multipoles in the multipolar expansion of cosmic microwave background temperature data from the Wilkinson Microwave Anisotropy Probe (WMAP), recently reported in the literature. We introduce an alternate statistic which probes this effect more sensitively. We find that the data are highly anomalous and consistently outside the 2σ significance level in the whole multipole range l = [2, 101]. We examine the possibility that this asymmetry may be caused by the foreground cleaning procedure or by residual foregrounds. By direct simulations we rule out this possibility. We also examine several possible subdominant foregrounds, which might lead to such an asymmetry. However, in all cases we are unable to explain the signal seen in data. We next examine cleaned maps, using procedures other than that followed by the WMAP science team. In particular, we analysed the maps cleaned by the Internal Powar Spectrum Estimation (IPSE), needlets and the harmonic Internal Linear Combination (ILC) procedures. In all these cases, we also find a statistically significant signal of power asymmetry if the power spectrum is estimated from the masked sky. However, the significance level is found to be not as high as that in the case of the WMAP best-fitting power spectrum. Finally, we test the contribution of low-l multipoles to the observed power asymmetry. We find that if we eliminate the first six multipoles, l = [2, 7], the significance falls below the 2σ confidence level. Hence, we find that the signal gets dominant contribution from low-l modes.