Another idea that I have had, which I think can be another explanation of the Great Unsolved Mystery, why does Dumbledore trust Snape?
My previous ideas were: wizard debt to James Potter for saving Snape’s life, Snape in love with Lily Potter (still my number 1 choice.)
But here’s something new.
After Voldemort disappears from killing Harry Potter, why is Dumbledore so convinced that he will return? Seemingly alone in the wizard world - in chapter 1, Philosopher’s Stone, all the other wizards are celebrating!
Dumbledore knows that Voldemort is not dead, and will be back, because Voldemort has a horcrux (or more than one). How does he know this? Dumbledore might have suspected that Voldemort is making horcruxes (but it would not be his first thought, since Dumbledore knows very well that their is nothing at Hogwarts that even tells you what a horcrux is, let alone how to create one), or … Albus might have been told by Severus Snape! Snape, overcome with remorse after finding out that Voldemort will kill the Potters (and here I can include the Snape loves Lily theme into my idea), goes to Dumbledore and tells him “Voldemort is unkillable, he knows how to make a horcrux, and already has one! He got the idea from blabbermouth Horace Slughorn!”
Dumbledore knows that this is something that Voldemort would never want to be known. This is a major proof of Snape’s loyalty.
Another proof: in Half-Blood Prince, when Bellatrix Lestrange and Narcissa Malfoy visit Snape in his home, Snape delivers this information: “Dumbledore has been a great wizard, but the duel against Voldemort has much weakened him.” Another proof of his loyalty to Dumbledore! He lied here - Dumbledore is not weak because of the duel against Voldemort, Dumbledore is weak because he was wounded when destroying the horcrux hidden in Marvolo’s ring! Snape and Dumbledore know that news of Dumbledore’s injured hand will get to Voldemort sooner or later, so they concoct this cover story, which will be all the more believable to Voldemort because it will flatter his vanity (I knew that I could beat Dumbledore in a fair fight if no one interrupted!)
I can even throw in an answer to the question “Why does Dumbledore trust Snape to teach DADA in Half-Blood Prince, when he never trusted him before?” Becuase Snape helped him destroy a Voldemort horcrux (the one in Marvolo’s ring). We know, since Dumbledore told Harry, that the injury to Dumbledore would have been fatal if Snape had not helped him. Is it a big leap to assume that Snape was actually present? No!
Then the question arises - why would Dumbledore not bring Snape along with himself and Harry when they go get Slytherin’s locket in the cave? Several possibilities: Dumbledore needs Snape to remain behind to guard Hogwarts, because Dumbledore knows that Draco Malfoy is dangerous. And/or, Dumbledore was intending on bringing Snape along, but Harry bursts into the office shouting “Snape betrayed my parents!” Dumbledore becomes pale, because he knows that Harry now will never trust Snape, and that it would be a horrible idea to bring Snape along on their quest, because Harry and Snape will be fighting the whole time.