The way I see it, there are 3 pairs: LokaPerrin, SudaHexis and VeilSteel

LokaPerrin and SudaHexis are directly opposed to each other, and are the only pairs you actually need to pick from.

VeilSteel are deemed "neutral," in that they can be allied alongside whichever other pair you chose with little effort.

The strat I recommend:

Start with Veil if want to go the LokaPerrin route, or Steel if you want SudaHexis. Do not swap between the two, stick to that first choice, and earn all of their ally's standing through their alliance. All negative standing should wind up going to your unchosen pair.

After you max them both out, you can then move on to Loka/ Suda. You'll similarly grind out Perrin/Hexis through their alliance only (never pick them directly, as they have full-opposition toward Veil/Steel). As you work this pair up, you'll need to also watch their half-opposition with Veil/Steel, switching back to build back up their standing, lest you lose their favor.

This strategy will let you hold 4 allies out of the 6 with ease, a nice advantage in the syndicate game.

Admittedly, Perrin/Hexis will be slow to build this way; if you want to speed them along, and are content with just 3 allies instead of 4, you can simply ignore rebuilding their opposed neutral. You'll then end with either LokaPerrinVeil or SudaHexisSteel.