If I score both battlefields on my turn, then on my opponent's turn I move my unit from one battlefield to another (previously scored) battlefield, can I score that battlefield again on my opponent's turn?

Ruling: Yes, you can score the same battlefield again on your opponent's turn. The "once per Battlefield per turn" restriction resets each turn, regardless of whose turn it is.

Sequence:

• Your turn: You score Battlefield A and Battlefield B (2 points total), end turn
• Opponent's turn: Opponent starts a showdown at Battlefield A
• During the showdown: You move your unit from A to B
• Battlefield B becomes Contested (if it was empty/uncontrolled)
• You establish control/conquer Battlefield B on opponent's turn (1 additional point)
• Total: 3 points (2 from your turn, 1 from opponent's turn)

Nuances:

• The scoring restriction is "once per battlefield per turn" where "turn" means any player's turn, not just your own turn
• Each new turn (whether yours or opponent's) resets the scoring restriction for all battlefields
• You can only score if you actually establish control through conquering - simply maintaining control doesn't trigger scoring
• If you leave a battlefield and don't return to it during opponent's turn, you don't get another scoring opportunity there