A Southern Auroral Electrojet (SAE) index has been recently constructed using several Antarctica magnetometer stations. It has been compared for case studies with the standard Auroral Electrojet (AE) index, and a near-conjugate to the southern stations Northern Auroral Electrojet (NAE) index. We compare the three indices statistically as a function of the accompanying solar wind (SW) and Interplanetary Magnetic Field (IMF) conditions to further explore conjugacy issues. We use 274 days of common north/south data presence between December 2005 and August 2010. We calculate the cross-correlation coefficients and differences between all three pairs. We estimate the effect of the SW/IMF conditions on the index correlations and differences using three groups of data: (a) the entire data set, (b) two separate sets based on the presence or not of Southern Hemisphere stations within the 21-03 Magnetic Local Time (MLT) sector where substorms occur, and (c) separately for the four different seasons. We find that high north-south correlation coefficients are more common during strong SW/IMF driving, while the index differences are also higher, suggesting that the SAE index follows better the northern indices' trend, but has even lower values during active times. The UT study shows that the number of high AE/SAE correlations is slightly lower at all clock angles and dynamic pressure levels for the periods within 1454-1941 UT (when no southern station is within 21-03 MLT). Finally, the results show that the number of high correlations is greater during the northern spring than the winter period. The Auroral Electrojet (AE) index is a measure of geomagnetic activity, resulting from the interaction between the solar wind emanating from the Sun (and the accompanying solar magnetic field) and the Earth's magnetized environment, the magnetosphere. The standard AE index is measured in the Northern Hemisphere auroral zone. We examine the agreement between the recently constructed Southern AE index (SAE), the standard AE index, and a magnetic equivalent to SAE, Northern AE index (NAE). The comparison is conducted in conjunction with various Solar Wind (SW) and Interplanetary Magnetic Field (IMF) properties, for three data sets: (a) the entire SAE data set, (b) two separate sets based on the presence or not of Southern Hemisphere stations within the midnight sector where the most severe geomagnetic disturbances occur, and (c) separately for the four different seasons. The entire SAE data set reveals that for active periods SAE follows the trends of the northern indices, but at the same time exhibits higher separation in value from them. Our second study shows that caution should be used for times when no southern station is within the midnight sector. Finally, more consistent north/south indices occur during the northern spring season in comparison with northern winter. The correlation between AE/Northern Auroral Electrojet (NAE) and Southern Auroral Electrojet (SAE) increases with solar wind driving, while the magnitude of the AE/NAE indices is higher than the SAE The North-South symmetry of AE indices is modulated by Interplanetary Magnetic Field By Northern spring has higher correlations and differences compared to northern winter