If an opponent moves into a battlefield I control, can I activate a Hidden Tideturner on another battlefield I control to swap units?

Yes, you can play a hidden Tideturner from another battlefield you control to swap units with the battlefield where the showdown is occurring.

Legal Basis

According to FAQ 1652 and FAQ 10693, you are permitted to play reactions from any battlefield when there is an item on the chain, regardless of where the showdown is taking place. Furthermore, Tideturner possesses specific errata (as clarified in FAQ 6517, FAQ 7907, and FAQ 2851) that allows its "When you play me" triggered ability to target a unit at any location, bypassing the standard restriction that cards played from hidden must target units at their own battlefield.

Sequence of Events

1. Showdown Initiation: An opponent initiates a showdown. To play a reaction, you must wait until an item (such as an attack trigger or an action-speed effect) is placed on the chain, or until the opponent passes focus and priority.
2. Reacting: The turn is in a Closed State because an item is on the chain. You play your hidden Tideturner from your other battlefield as a [Reaction].
3. Entry: Tideturner enters the board at its original location. Per FAQ 6244, playing a permanent from hidden starts a chain that resolves immediately (you cannot react to the unit entering play itself).
4. Triggered Ability: After Tideturner is in play, its "When you play me" ability is placed on the chain.
5. Targeting: Because of its specific errata, you choose your unit at the attacked battlefield as the target for this ability.
6. Resolution: The ability resolves, swapping the positions of Tideturner and your original unit. Your original unit moves to your other battlefield, and Tideturner moves to the battlefield where the showdown is occurring.

Note: I could not fully verify this ruling against the retrieved rules.