What I may have learned about this data is that conventional time series analysis will drive you crazy.
The SOI data has to be transformed somehow so that it starts to show the underlying periodic nature. Something as simple as adding the 2nd-derivative to the waveform itself will partially cancel the characteristic frequency and begin to reveal the details.

As a case in point, Jim's three periodograms do not show the 2.33 year QBO frequency but which are readily revealed via comment #43. So the key is trying to re-evaluate the data in different ways, and definitely wavelets are part of that strategy.

And again as I mentioned before, I am looking for a metric that can correlate between wavelet scalograms. It appears that this gives another degree of freedom, i.e. 2 axis instead of 1, so that it will be much more difficult to obtain fortuitous correlations, which is a common problem of time series analysis, especially with overfitted models. Yet in two dimensions, no two fingerprints are alike so that it might help to substantiate the agreement between model and data.