How to split climate modes of variability into positive, negative and neutral phases?
I think thats a lot of information on the link you pasted. and the indexes shown there are mostly in time (time-series). 
Please note two important steps. 
FIRST: the index you want to use is already produced in scientific way and published so how is the index produce please read the relevant paper for that which can help. Apart from that these indexes are mostly (Not all of them) are usually developed to capture/shows standardized/normalized/Anomalous  behaviors of the system. So these indexes are representation of the systems with mean of zero and standard deviation of usually one in case of standardized index, and so does varies for each one of them. The anomaly is different story and i won't go there.

SECOND: Now the deviation of the index from mean (which is zero) shows the deviation from mean conditions (that means from neutral conditions). So positive and negative will show the modes of system based on intended interpretation. like for strength positive deviation will be termed as strong and vice versa. 

Finally you can select the positive and negative years as modes of climate with positive and negative deviation from mean. Further to dig deep you can look for mechanism and principals that led to these deviations. 

If you want to use your data and look for modes of climate, then i will suggest PCA, SVD and EOF analysis which may not be applicable in this case.
I hope that will help