To seize the corner of a man’s garment was, in the ancient world, to pronounce loyalty and dependence on the man. We saw this in chapter 15 when Saul seized the hem of Samuel’s robe. Unfortunately, Samuel’s robe tore, which led to the very opposite idea: pronounced rebellion. David’s cutting of Saul’s robe indicated the division between them. This is why David is so remorseful later: he has with his actions openly rebelled against God’s anointed.
The chapter is full of symbolism like this.
In calling Saul his “father,” there is implied claim to the throne of Saul. Saul may consider David a “dead dog” or a “flea,” but in David’s mind, on God’s side, Saul has vastly underestimated David, particularly because David is claiming to be on the Lord’s side and Saul has, by his actions, taken on God. David is willing to leave it up to God, but he’s not just saying: “We’ll leave it up to God to decide.” David is actually calling on God to hurt Saul, to judge him, to avenge David, and to deliver him from Saul’s hand.
And surprisingly (though he doesn’t mean it), Saul asks that God reward David. David has cut off the hem of Saul’s robe. But Saul requests that David not “cut off” Saul’s descendants.
Centuries later, Jesus will say: “Love your enemies. Do good to those who hate you” (Luke 6:27). The story of David is an example of just such submission to the will of God. David wants Saul to suffer for the way he has wronged him, but he unwilling to make Saul suffer himself. He leaves it up to God to decide and act. It is, in fact, the way of Jesus, and should be the way of all his disciples.