Leviticus 11:27 Of all the animals that walk on all fours, those that walk on their paws are unclean for you; whoever touches their carcasses will be unclean till evening.

So, dog is unclean.

Deuteronomy 14:8 The pig is also unclean; although it has a divided hoof, it does not chew the cud. You are not to eat their meat or touch their carcasses.

What is the correct interpretation of Matthew 7:6?

The answer can be found in 2 Peter 2.

New Living Translation 2 Peter 2:1 But there were also false prophets in Israel, just as there will be false teachers among you. They will cleverly teach destructive heresies and even deny the Master who bought them. In this way, they will bring sudden destruction on themselves. 2 Many will follow their evil teaching and shameful immorality. And because of these teachers, the way of truth will be slandered. ...
21 It would be better if they had never known the way to righteousness than to know it and then reject the command they were given to live a holy life. 22 They prove the truth of this proverb: “A dog returns to its vomit.” And another says, “A washed pig returns to the mud.”

These false prophets and teachers were once cleansed by the righteous truth but then made themselves unclean again by corrupting the very truth that they were once taught.

Matthew 7:6 Do not give dogs what is holy; do not throw your pearls before swine. If you do, they may trample them under their feet, and then turn and tear you to pieces.

In Matthew 7:6, Jesus prophesied the coming false Christians leaders who would destroy their followers.

Leviticus 11:27 Of all the animals that walk on all fours, those that walk on their paws are unclean for you; whoever touches their carcasses will be unclean till evening.

So, dog is unclean.

Deuteronomy 14:8 The pig is also unclean; although it has a divided hoof, it does not chew the cud. You are not to eat their meat or touch their carcasses.

What is the correct interpretation of Matthew 7:6?

The answer can be found in 2 Peter 2.

New Living Translation 2 Peter 2:1 But there were also false prophets in Israel, just as there will be false teachers among you. They will cleverly teach destructive heresies and even deny the Master who bought them. In this way, they will bring sudden destruction on themselves. 2 Many will follow their evil teaching and shameful immorality. And because of these teachers, the way of truth will be slandered. ...
21 It would be better if they had never known the way to righteousness than to know it and then reject the command they were given to live a holy life. 22 They prove the truth of this proverb: “A dog returns to its vomit.” And another says, “A washed pig returns to the mud.”

These false prophets and teachers were once cleansed by the righteous truth but then made themselves unclean again by corrupting the very truth that they were once taught.

Matthew 7:6 Do not give dogs what is holy; do not throw your pearls before swine. If you do, they may trample them under their feet, and then turn and tear you to pieces.

In Matthew 7:6, Jesus makes two points.

1. Don't waste your breath on certain unholy people. They may just ignore you or worse,
2. they may misconstrue what the truth is and become false Christians leaders who would destroy their followers.

Matthew 7:6 can be interpreted as a prophecy.