Why did Jesus use spit to heal a blind man?

In all societies, human spittle is disgusting, contaminated and germ-laden. Spitting on someone is a grave insult and shows that you consider him despicable. When someone is spat upon you can be assured that he will become very angry.

John 9:1-7
¹As he went along, he saw a man blind from birth. ²His disciples asked him, "Rabbi, who sinned, this man or his parents, that he was born blind?"
 ³"Neither this man nor his parents sinned," said Jesus, "but this happened so that the work of God might be displayed in his life. ⁴As long as it is day, we must do the work of him who sent me. Night is coming, when no one can work. ⁵While I am in the world, I am the light of the world."
 ⁶Having said this, he spit on the ground, made some mud with the saliva, and put it on the man's eyes. ⁷"Go," he told him, "wash in the Pool of Siloam" (this word means Sent). So the man went and washed, and came home seeing.

I have heard some strange theological explanations: that it was a symbolism of the creation time in Genesis, that it was a medicinal (huh?) poultice.

For a clue to the answer, have you wondered why he insulted this Samaritan woman who asked him to heal her son?

Matthew 15:21-28
²¹Leaving that place, Jesus withdrew to the region of Tyre and Sidon. ²²A Canaanite woman from that vicinity came to him, crying out, "Lord, Son of David, have mercy on me! My daughter is suffering terribly from demon-possession."
 ²³Jesus did not answer a word. So his disciples came to him and urged him, "Send her away, for she keeps crying out after us."
 ²⁴He answered, "I was sent only to the lost sheep of Israel."
 ²⁵The woman came and knelt before him. "Lord, help me!" she said.
 ²⁶He replied, "It is not right to take the children's bread and toss it to their dogs."
 ²⁷"Yes, Lord," she said, "but even the dogs eat the crumbs that fall from their masters' table."
 ²⁸Then Jesus answered, "Woman, you have great faith! Your request is granted." And her daughter was healed from that very hour.

A dog in eastern cultures is despised and unclean. So why did Jesus insult her?
The answer is simple: Jesus was asking every one of them,

"Are you willing to give up your dignity in exchange for a miracle?"

Notice that they had a choice to feel insulted and leave, or to remain, press closer and obey. This is also seen in an Old Testament healing where the prophet Elisha asks Namaan the general of Syria who had leprosy to wash in the lowly, muddy Jordan river - not once, but seven times. Again the question:

"Are you willing to give up your dignity in exchange for a miracle?"

Finally, Jesus himself was faced with that question.

"Are you willing to give up your dignity in exchange for a miracle?"

Jesus was spat upon, beaten, mocked, insulted, whipped, and crucified. His final dignity was removed when he hung naked on the cross. He did this in exchange for the miracle of salvation for you, and for me.