Can my opponent attack my battlefield where I have a unit, and can I react with Hidden Tideturner at the other battlefield to swap places with the first unit?

Yes, you can react to your opponent’s attack by playing a hidden Tideturner from another battlefield to swap places with your unit at the attacked battlefield.

Legal Basis

According to FAQ 1652, you can play reactions from any battlefield when there is an item on the chain, regardless of where the showdown is occurring. Furthermore, Tideturner has specific errata (as noted in FAQ 6517 and FAQ 2851) that allows it to bypass the standard restriction that hidden cards must target units at their own battlefield. Because Tideturner's ability explicitly requires targeting a unit at another location, it is permitted to choose targets freely across battlefields.

Sequence of Events

1. Showdown Trigger: Your opponent moves a unit to your battlefield, initiating a showdown.
2. Reacting: You play your hidden Tideturner from your other battlefield in response to the attack trigger on the chain.
3. Entry and Ability: Tideturner enters play at the battlefield where it was hidden. Its "When you play me" ability is placed on the chain.
4. Targeting: Because of its specific errata, you choose your unit at the attacked battlefield as the target for Tideturner’s ability.
5. Resolution: The ability resolves, swapping the positions of Tideturner and your original unit. Your original unit is now at your other battlefield, and Tideturner is now at the battlefield where the showdown is occurring.

Note: I could not fully verify the interaction between the 'CLOSED' state and the attack trigger against the retrieved rules.