So how do we pair moves up in bridge it (even without considering winning)?

[The blue dots in the figure below represent the available moves, which becomes unavailable after whoever move there first. So you want to pair the blue dots]

There are an odd number of moves, so remove a blue dot (which represents 1st player’s first move) before beginning to pair up dots.

If we now remove all the green and red dots, can you visualize a way to play the game involving blue dots only?

Here is how I would (the 1st figure on the right):
Consider the green, red and black bridges as part of the set up and the objective is the same, except that the blue area is unsafe to cross until it is claimed by green or red. Black bridges can be used by either player.

Since the center dot is heavily connected, it may be good for the first move.

Let’s now focus on green’s setup (assuming that green moves second. See 2nd figure) and red wants to find a pairing to block green. How should he pick? Try different ways to see what works.

Adjacent blue dots are good candidates to pair up since this blocks one of possible paths. Also, it’s better to pair up horizontally, since this is where green wants to go. If you pair one diagonally, the rest of the diagonal pair near it should be in that direction because that keeps the opponent from moving forward (the figures below show negatively slopping diagonal pairs).

The figures show two methods to pair.

Bigger boards are also shown (fill in the rest).

You may prefer the second, which uses the corner for the first move instead.

Exercise: Do the pairings work?
How does the pairing (horizontal and diagonal) correspond to the tactics in 4c?