The purpose of the Desaspected Crystals is to exchange them for wither shards or a Perigrine Helm
. You need 30 for the helm and considering you seems to get around 5-8 each time you would need to fight 5-6 invasions.
The HQ gear drops from the mobs themselves. I partied with some LS mates and a bunch of randoms taking part and a Cotton Cowl +3 dropped into someone's loot list at one point. After you are successful there will be a notice flash up that the Garleans have retreated, then another saying that a treasure chest has spawned. You just need to run back to the guy with the chocobo to find the chest and get your crystals.
The other 'reward' so to speak is an accomplishment on your Lodestone History page (if you are into reaching accomplishments). There are 3 you can get:
Empire Strikes back - for talking to Alfgar or Roysia when they appear and taling through each option
A Taste of Victory - for the first time you take part in and defeat a Garlean Force
Rhapsody in White - for exchanging crystals for the Helm