I have shown before how climate response to volcanic forcing varies with thermal inertia. A volcano catching the oceans on a downward trend has more impact than catching it on an upward trend. The same thing with solar and ENSO. It takes time for equatorial heat absorbed in the oceans to travel toward the poles. When the ENSO "cycle" shifts, the temperature differential will be different between the poles and equator producing a different response. This is due to a combination of real inertia, the mass of the ocean currents combined with the heat contained in that mass.
The chart above shows the same situation with solar. Since both solar and ENSO have different factors that determine their "cycles" they can become in phase or out of phase producing a different response to the same amount of "forcing". In 1955, the two were close to 100% out of phase and by 1988-89 they were close to 100% in phase. Since the ENSO period is shorter, it tends to "breakdown" faster than the solar "cycle".
The use of "cycle" in quotes is because they are cycles, but due to different interactions they appear to be pseudo-cycles. That is just part of dealing with the real, messy world of non-linear systems.
Because of the number of influences involved, determining a reliable impact per perturbation is nearly impossible. The ocean alone has probably 15 layers or segments that would need to be considered which are of course asymmetrical relative to the equator with all that atmospheric "weather" feeding back and forward on the ocean "cycles" at various depths. It is chaotic.
How you "prove" the impact of phasing of solar/ENSO or ENSO/AMO etc. is still a big question mark. Getting people to realize that there is a definitive impact is hard enough.
Just food for thought.