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Bug

No, it is not correct:

self.right.set_height

set_height is not run due to the lack of parenthesis, hence the height is not correctly set.

Swapping

In many places such as for example:

    temp = self
    self = self.right
    self.right = temp
    self.right.set_height
    self.set_height

You are swapping values in an obfuscated manner, just use the built-in a, b = b, a syntax:

>>> a, b = 1, 2
>>> a, b = b, a
>>> a, b
(2, 1)

Repetition Repetition

There must be a way to avoid repetition in the below code-blocks, the problem is that you have duplicate functions for left and right, maybe a single rotate function could help you here.

        '''Adding to left subtree'''
        if self.left is None:
            self.left = Node(data)
        else:
            self.left.add_node(data)
            if ( self.get_left_height() - self.get_right_height() == 2):
                ''' Then we need to balance a subtree'''
                print("Rebalancing after inserting", data)
                if (data < self.left.data):
                    self.rotate_left()
                else:
                    self.double_rotate_left()

        '''Adding to right subtree'''
        if self.right is None:
            self.right = Node(data)
        else:
            self.right.add_node(data)
            if ( self.get_right_height() - self.get_left_height() == 2):
                ''' Then we need to balance a subtree'''
                print("Rebalancing after inserting", data)
                if (data < self.right.data):
                    self.rotate_right()
                else:
                    self.double_rotate_right()