What cadence saves power when climbing? I'm doing some amateur climbing with a hybrid bike up on a road and I have the following question:

How am I going to save more power - having a lower gear but spinning faster in order to maintain certain speed or getting up to a higher gear but spinning slower to aim for that same speed?

I would love some feedback as I'm experimenting in the mountains but every day is different in regards to moisture, temperature, sun exposure, nutrition in the morning, etc. and at this point it's hard to make a good conclusion so I'm looking for some science to help with this question.
Honestly, as an amateur cyclist I would suggest focusing on learning how to pace a climb first over more finer details such as cadence.  Many amateur typically go out too fast on a climb, go anaerobic, accumulate a lactate debt, then find themselves suffering terribly the remainder of the climb.  This gives most the idea the climbs are harder than they could be. Ideal, you want start easier, settle into a sustainable pace, and only max out your effort at the top of the climb.  This can take practice.  I would err on the side of going too easy for the whole climb.  See how that feels. Next time you do the climb try upping the effort a bit.  Keep doing this until you find yourself blowing up on the climb... congratulations you found your aerobic threshold. Next time you will know to back off a bit and this will be pretty much your max sustainable pace.

In terms of cadence, try different gears until you find one that feels good.  When left to their own devices people tend to self select a suitable cadence for their training and muscle coordination. (I am sure there are exceptions to this, but once people learn how to change gears many do a good job of self-selecting an appropriate cadence.)

Higher cadence climbing tends to be a technique to "save" your legs for later hard efforts like sprinting (i.e., you are trying to avoid tapping your fast twitch muscle fibres on the climb, these fibres are what drive short hard efforts).  This trade-off can be important if you are racing, less so if you are a causal rider. (Unless you are riding with friends and there are fish signs sprints - it can be important to be at your best so you can win all the beer!)

Anyway, I digress... Fast cadence climbing is a technique that has to be trained. If you do all your riding at a slow cadence, you will not have efficient muscle coordination for climbing at a fast cadence.  I personally found things like fast paced club rides a natural place to hone the fast cadence riding style.  Training it on your own might be a bit tedious and boring, but can be done.  

Strangely, enough lower cadence climbing may actually be more efficient in terms of oxygen consumption (i.e., cardio-respiratory system work; see Determinants of “optimal” cadence during cycling). This is consistent with the idea of "saving" your legs, where you are stressing your cardiovascular system a bit more in order to try and reduce some of the stress on the muscular system.