The Basics
Swirl is a reaction that can only be caused by an Anemo user as the reactor ( Unless you're fighting an Anemo slime or Hypostasis ), meaning that you can't apply an Anemo aura to an enemy and have another character apply an element to cause Swirl. Anemo can only react with an existing elemental aura to cause Swirl ( Hydro / Pyro / Electro / Cryo ). This means that the Anemo user's Elemental Mastery and Character Level will always be used to calculate Swirl damage. Anemo cannot react with Geo.
Swirl's base reaction causes the Swirled element to be spread to enemies in an AoE. For Pyro / Electro / Cryo, this causes AoE damage ( of the respective element ) and the elemental aura is applied to nearby enemies. Hydro is an exception, in that it only causes the Swirled enemy to be damaged in single target, while spreading Wet to nearby enemies - it doesn't do AoE damage. If you are Swirling in AoE, it is possible to keep an elemental aura up on enemies indefinitely by continually Swirling the element back and forth. Swirling the same enemy repeatedly in single target will naturally cause the elemental aura to eventually expire.
Chain Reaction
The part that really makes Swirl powerful is the potential for chain reactions. Swirl can cause subsequent elemental reactions when a Swirl reaction of an element hits another enemy with a different elemental aura, and the resulting reaction's damage will be calculated based on the Anemo user's Elemental Mastery and Character Level.
For example, if you have two enemies, and one has a Pyro aura and the other has an Electro aura, Swirling them with an AoE ability will cause both enemies to react with Overload.
Venti Swirling Pyro + Electro to cause Overload, his Elemental Mastery and Character Level determine the damage
Similarly, if you have multiple enemies, each with different elemental auras, Swirling them all together using an AoE Anemo ability will cause them all to react with each other.
AoE Limit
Swirl has amazing AoE potential because of this. However, there is a limit you must account for. Swirl can only cause two instances of AoE damage per element for a single AoE ability. This means that if you have four enemies with Electro auras, and Swirl them with an AoE ability, only two instances of Swirl Electro damage will occur per enemy. Hydro again is the exception here, as it doesn't do AoE damage, and so there is no limit on it - you can Swirl seven enemies that each have a Hydro aura and see seven Swirl instances of single target damage.
On the other hand, if you have multiple enemies with multiple auras, you can Swirl up to two times per element and generate a lot of AoE damage. With chain reactions, you could cause Overload on every enemy for example.
This can easily be put to use on most Abyss floors, as there is generally at least one enemy with an elemental aura already on them, or some floor effect constantly applying an elemental aura. You only need to supply another element to cause a Swirl chain reaction. Off field elemental appliers like Fischl, Beidou, Lisa, Xiangling, Xinyan, Barbara, Xingqiu, Mona, Diona and Ganyu can make things easier. They can allow you to persistently apply your own auras and Swirl them together for more reactions.
Notably Swirl is extremely useful on Abyss 12-1-1, where once the enemies are infused with Pyro and Electro they can be constantly Swirled into each other to spam Overload.
The individual numbers of Swirl are low, which causes it to be typically underrated. However, Swirl plus its chain reactions can be spammed in AoE effectively to greatly increase your DPS on multiple enemies.
Electro-charged
Electro-charged is typically the best chain reaction to use for consistent damage ( and stagger crowd control! ). The Anemo user's Character Level and Elemental Mastery will be of higher value with this chain reaction. This is because Electro-charged creates a dual-aura of Electro and Hydro, and Anemo will cause both a Hydro and Electro Swirl at the same time - effectively spreading the reaction. If you are using an Anemo user with an Elemental Burst with an absorption mechanic ( Venti, Sucrose, or Anemo Traveler ) absorbing Hydro into their Burst with an enemy that has an Electro aura is the easiest way to create a self-sustaining Electro-charged Swirl reaction. This is because the Anemo portion of the Elemental Burst will cause Electro Swirls to spread Electro, and the absorbed Hydro portion of the damage will then proc Electro-charged continuously.
Freeze
Freeze is the best reaction to use for consistent crowd control, although the damage output for Swirls will be much lower. The value of Character Level and Elemental Mastery to your Anemo user is much lower if you are using Freeze as a chain reaction. This is because although Freeze involves Cryo and Hydro, both cannot be consistently Swirled at the same time like with Electro-charged.
The chain reaction of Freeze also naturally does not cause any damage, and if you are Swirling Hydro it will only do single target damage per Swirl. Outside of the specific cases previously mentioned, Anemo damage will cause either a Hydro or Cryo Swirl on a Frozen enemy, the element that gets Swirled seems to depend on the strength and order of the Hydro + Cryo application.
Melt and Vaporize
Melt and Vaporize will multiply off of Swirl's base damage. Melt and Vaporise reactions will typically end up consuming one element over continuous Swirls and result in a mono-element spread across all enemies. Swirling elements for these reactions is typically used to help keep up auras for other party members to consistently react with.
4-piece Viridescent Venerer
This artifact set improves Swirl damage by 60%, but also decreases an enemy's elemental resistance to the element infused in the Swirl by 40%, further increasing the damage output of Swirls. While it functions very well for supporting other party members, this set is also very important to getting the most out of your Swirl damage. An important factor to note is that the debuff from the set will only be applied if your Anemo user is on field at the time of the Swirl reaction.
In the case of a double Swirl from Electro-charged or Freeze, both elemental resistances will be debuffed on the enemy.
Notable Interaction
Anemo users being forced to be the primary reactor can be taken advantage of, not only by utilizing Elemental Mastery but also in situations like with Fischl's ascension4 passive. Fischl's passive states that if your active character triggers an Electro-related Elemental Reaction when Oz is on the field, she will deal Electro DMG equal to 80% of her ATK. Since Anemo users always have to be the reactor, and you can easily keep up an Electro aura on enemies by Swirling in AoE, constantly proccing Fischl's ascension4 passive for a massive DPS increase is quite easy ( Geo users are also great at utilizing this passive for the same reason ). If you cause a chain reaction Electro-charged, this will also proc Fischl's passive.