I was searching for any references to a strict biennial == 2-year periodicity in the climate science literature and found  by Kim in 2003. This doesn't have many cites other than by the author to his own work, but it may hold the key.
From the abstract:
"The first CSEOF is stochastic in nature and represents a standing mode of SST variability associated with a basinwide change in the surface wind. The second CSEOF exhibits a strong deterministic component describing a biennial oscillation between a warm phase and a cold phase. The surface wind directional change in the far-western Pacific appears to be instrumental for the oscillation between the two phases. Because of the distinct nature of evolution, dynamical and thermodynamical responses of the two modes are different. Further, the predictability of the two modes is different. Specifically, the biennial mode is more predictable because of the strong deterministic component associated with its evolution. The distinction of the two modes, therefore, may be important for predicting ENSO. The irregular interplay of the two modes seems to explain some inter-ENSO variability, namely, variable duration of ENSO events, approximate phase-locking property, and irregular onset and termination times."
 Kim, Kwang-Yul, James J. O'Brien, and Albert I. Barcilon. "The principal physical modes of variability over the tropical Pacific." Earth Interactions 7.3 (2003): 1-32. [PDF](http://journals.ametsoc.org/doi/pdf/10.1175/1087-3562%282003%29007%3C0001%3ATPPMOV%3E2.0.CO%3B2)
Here is a selected passage where they describe the even and odd year effects
"As they propagate eastward, the upwelling (negative) Kelvin waves erode the thermocline structure favorable for the maintenance of the positive sea surface temperature anomaly condition in the central and eastern Pacific. This terminates the warm phase of the biennial oscillation. It takes about 3 months for the Kelvin waves to reach the eastern boundary; then, the colder SST condition is observed in the eastern Pacific. In the winters of even years, the situation reverses. An easterly surface wind anomaly is observed in the far-western Pacific, and henceforth the thermocline deepens. The warmer subsurface water at the thermocline depth propagates eastward as downwelling Kelvin waves as the direction of the surface wind anomaly changes in the western Pacific by the end of the even year. Then, a warmer surface condition is observed in the following spring"
This seems like a compelling story, but why isn't there more on this topic of even versus odd year ENSO effects? It is possible that since this was published in 2003, that something happened to cause the even/odd years to switch places and so scientists reading the paper later would think it was not applicable. It is difficult for me to see how an odd year is special over an even year, and so one should be open to the idea that it could reverse.
Notice also the reference to the *"about 3 months for the Kelvin waves to reach the eastern boundary"*, which happens to be the delay from Sydney harbor tidal readings to the corresponding SOI readings that I discovered in the correlation.
There are many reference to biennial variations in Pacific Ocean salmon catch -- google Salmon "Odd Year" "Even Year".
In recent times , the salmon catch has been higher in odd years, but this has flipped in the past.
 Irvine, J. R., et al. "Increasing Dominance of Odd-Year Returning Pink Salmon." Transactions of the American Fisheries Society 143.4 (2014):
939-956. [PDF here](http://www.tandfonline.com/doi/full/10.1080/00028487.2014.889747#.VBsx07RtR8w)