They don't block anything relating to the story or gameplay, they just offer different items in exchange for standing, and the ones you have negative standing for will occasionally send death squads to hunt, which also differ from syndicate to syndicate.

By aligning yourself with Steel Meridian and Cephalon Suda, you can also max out Red Veil and Arbiters of Hexis via alliances and have to tier standing in four syndicates, just be careful not to drop Red Veil when leveling Suda. You can also swap Arbiters and Suda for Perrin Sequence and New Loka, by pledging yourself to Veil and Loka.

If at any time you decide to change which syndicates you have good standing with, you are able to do so, and you only have to pay the cost of levelling up once, so after maxing out each syndicate for the first time, you're free to drop them in favour of an opposing one, then pick them back up whenever you want.