If I have scored on both battlefields this turn, move off my units and then move a unit onto the battlefield, does a showdown start?

Yes, a showdown does start when you move a unit onto that battlefield, even if you have already scored on both battlefields this turn.

According to RiftJudge FAQ #10027 and Rule 344, a showdown occurs whenever a battlefield becomes contested, regardless of whether it has already been scored that turn.

Here is the breakdown of why this happens:

1. Losing Control: When you move your last unit out of the battlefield to your base, the battlefield becomes unoccupied and uncontrolled (Rule 323.6).
2. Contesting: When you move a unit back into that battlefield, you are moving into a location that you do not currently control. This makes the battlefield Contested (Rule 445).
3. Showdown Trigger: Because the battlefield is contested and the turn is in a neutral open state, a showdown begins (Rule 344).

Important Nuances:

• No Scoring: You will not score another holding point or trigger conquering effects (such as from specific cards) at that battlefield for the rest of the turn, because you have already scored it earlier (Rule 465, FAQ #7447).
• Non-Combat: Because the battlefield was unoccupied when the showdown began, this will be a non-combat showdown (unless your opponent also moves a unit there simultaneously, creating a combat scenario). Per Rule 348.2.a, since you are the only player with units present, you will regain control of the battlefield once the showdown closes.