RAW
It appears to be two actions, honestly much to my surprise.
Move Action
You can take a move action in place of a standard action.
So when you do that, you get two move actions. You can use both to move up to your speed, as separate actions.
Nowhere under full-round actions is any mention made of a double-move action. Running and charging, yes, but not the simple double-move.
Now, Swarm Attack reads as follows:
they deal automatic damage to any creature whose space they occupy at the end of their move (no attack roll needed).
I read “their move” here to be “their move action” rather than “their movement.” I do this because, grammatically, “move” is not really short for “movement,” while having an implicit noun for an adjective (i.e. action for adjective move) is quite common in English.
So yes, this would trigger Swarm Attack twice.
However, it’s always good to acknowledge ambiguities in the rules.
Should it be run this way?
I’m not sure. I’ve never seen it run that way, and I sort of doubt that most swarms were designed with that ability in mind, but I could be wrong; after all, I was wrong about this working in the first place. In any event, just because the authors intended it one way doesn’t mean that’s how I want to run it in my game.
For my games, I’d have to carefully consider the swarms I was or was not using, and what options my players had for dealing with them, in order to determine how I would rule this. In my experience, swarm damage tends to be low, but at the same time the majority of character classes have almost-zero ability to do anything to them.
So, for example, if my party consisted primarily of mundane classes, I really would not make swarms twice as dangerous as I thought they were, because I already considered them too dangerous by virtue of their effective invincibility. So, and I suppose it’s a houserule considering RAW, I would not allow swarms to double their swarm attack damage by double-moving.
On the other hand, if I was making a campaign centered around swarms, and expected my players to build specifically with swarms in mind, I probably would go ahead and rule this as RAW, and let swarms do that. This would allow the swarms to be more of a threat for situations where the players aren’t just enduring them, but are actively and effectively killing them. It would also open up a few more tactics for dealing with them, such as partial action denial, à la Nauseated.
If I anticipated swarms being an issue either way, I’d make a point of informing my players of this issue, since it’s unlikely that any of them have considered it on their own.
As for “verisimilitude,” or “seeing it through the game world’s lens” or whatever, that doesn’t really come into play here: swarms are explicitly an abstraction, particularly with respect to the Swarm Attack (using one lump sum rather than thousands of minuscule attacks), so it’s really impossible to say what makes more sense in-character; in-character, a completely different thing is happening.
The mount and the rider each have their own full turn with movement and action. When a player mounts an animal and decides to control it:
The initiative of a controlled mount changes to match yours when you mount it.
The mount's position in the turn order changes but it keeps its own turn on which:
It moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge.
Therefore the mount can move its full speed and finish its turn. After that the character takes his own turn on which it can dismount using half of the speed and move for the remaining half. What the character cannot do though is to mount, ride, and dismount as it would require two character's turns.
There is one thing that requires clarification though. If I understand correctly, ties in initiative are resolved once and for all, which means that after mounting the player has to decide whether it is the mount or the character who acts first during the round.
The only thing I see that may hurt my reasoning is this sentence (emphasis is mine)
A controlled mount can move and act even on the turn that you mount it.
It makes things arguable, but for me everything else is solid enough to consider that there is a mistake and it should be “round” instead of “turn” in this sentence.
Best Answer
So, I looked this up and here's what I've found. In a normal round you can take either one complex action (Attacks before unlocking the simple action attack option), two simple actions (drawing weapons, moving Water x 10 feet, etc.), and a reasonable number of free actions that I believe cannot be doubled.
Anyway, as for your question, it's perfectly reasonable to take two simple actions to 'double move' for a total of Water x 20 feet, which is the normal maximum movement. I would assume that complex move actions are for things like, I don't know, safely ducking under, leaping over, stepping between a spear gauntlet set up by a rival lord on the way to an objective (the L5R equivalent of a laser hallway). I don't see this coming up a lot though, so it's probably under the GM handwave territory.