Temples of Communion spread the Advent's culture into adjacent systems and beyond. The spread of culture is represented by the phase lanes connecting systems turning their color. This increases allegiance at planets owned by you and your allies, and decreases allegiance at enemy planets.
The Advent unlock the Temple of Communion after researching Unity Indoctrination. The Advent also have multiple upgrades that when used will improve the rate at which culture spreads as well as the bonuses received when fighting inside your own culture. Zealous Worship, Unwavering Belief, and Total Assimilation will improve Culture spread rate by 30%, shield mitigation by 6%, and give +20% phase missile block when in your own culture. Wall of Faith - At maximum level reduce the enemies resistance to your culture spread by 40%. False Belief Monitoring - Lets you see the spread of enemy culture anywhere in the Galaxy. Eyes of the Converted - Lets you see all planets that are under your culture.
Loyalist Advent have access to a lot of technology that will improve their culture. Confluence of the Unity - At max level increases culture spread rate by 60%. Acclimatization of Will - At max level increases the rate at which your culture lowers your enemies allegiance by 50%. Global Unity - Your planets will spread culture without the use of Temples of Communion. Fury of the Unity - Increases all damage in your culture by 20%.
Rebel Advent also have some technological improvements to their culture. Protection of the Unity - At max level Increases shield regeneration by 150% when inside your own culture. Mass Communion - Decreases the costs of Temples of Communion by 40%. Expulsion - Upon all enemies leaving your gravity well your planet enjoys a +25% increase to income and culture spread rate. Wail of the Sacrificed - While not directly affected by culture, it is an ability in the Temple of Communion that, when activated, sacrifices your planets population in order to do damage to all structures and ships in the adjacent gravity wells. It should be noted that while it earlier only affected enemy ships and structures it now does affect your own ship and structures.
Tactically the uses of culture is very limited as culture is very easily countered. However, this does not mean it is impossible. Early on it is wise to place a Temple of Communion down as this will do two things. Increase your allegiance of your planets, therefore giving you a small economical bonus. And force your enemies to fight in your culture as well as force them to make their own culture.
Also as of Rebellion, the Advent Rebels have acquired the Wail of the Sacrificed ability, which does a massive amount of damage to all units and structures should they be located in an adjacent planet. The damage is based off of population, so Terran and Desert planets will give you the most large amount of damage should your enemies not be careful and scout your planets. On a Terran planet with full upgrades towards Planetary Infrastructure and Inspired Architecture Wail will do 6440 damage to all ships and structures that are unlucky to get caught. This is a fleet wiping ability however it is generally only useful in a defensive position.