The minimum of solar cycle 23 exhibited anomalous characteristics with respect to its predecessors. Other than the differences in the standard indicators of activity, such as sunspots, polar magnetic fields were also found to be different. These differences had a range of effects in the solar wind. In this work we study the Alfvénicity and the residual energy of magnetic field fluctuations in two intervals from the solar minima in cycles 22 and 23 using Ulysses data. We find that the differences between the minima are mainly related to the greater prominence of convected structures in the broader equatorial region in cycle 23. Thus the total power in the fluctuations was lower in cycle 23, mainly due to a reduction in Alfvénic fluctuations. We also show that the relationship between the normalized residual energy and cross helicity is scale dependent.