You will find an optimal cadence for climbing as with riding on the flats. It is highly individual.
As for never changing gears, that is poor advice. If you watch an experienced rider start a climb, you will notice they often hit the base very fast. Carry your speed as long as possible. If you were flying on the flats, why shouldn't you get every bit you worked for?
Hit the climb and hold your "flat" speed as long as possible, while possibly "floating" (not working hard) a little. An object in motion tends to stay in motion!!!
As your speed drops, downshift until you settle into the climb. Strong climbers will usually shift to a harder gear the last portions of a climb.
Sometimes, a course or tactics will dictate that you hit the base slow, in the pack. I can think of one 12%+ climb on a local training ride where everyone goes into the small ring .5 mile before the base, only going 15-17 mph. Later, there is a one mile 6% grade that finshes the ride. We start slow in the big ring, usually a 53x21 or 53x19 at 13-15 mph until the attacks start, and the pace can hit 17-20 mph!!!
If you are going down a hill and facing another climb, I would spin out if possible in my biggest gear, and keep it there until I hit the base. I would then rapidly downshift while maintaining my cadence until I found the right gear. If you have to make the transition to your small ring, be sure your front derailleur is properly adjusted to avoid dropping your chain. Practice shifting on the fly and going between the small ring and big ring while climbing.
Good luck!!!