What does Luke 23:12 mean?

Herod Antipas, the son of Herod the Great, is of Arab and Edomite descent but nominally observes Judaism. He is the tetrarch of Galilee and Perea. Pilate is a Roman military officer and governor of Judea. Antipas is sympathetic to the Jews; Pilate's job is to keep order and collect taxes.

Dr. Darrell Bock, in his commentary Luke: 9:51—24:53, gives several potential reasons that Herod Antipas and Pilate might have been enemies.

Pilate hung shields in Herod's palace. The act displayed the fact that Antipas is subject to Rome, but the Jewish people appealed to Tiberius who ordered Pilate to move them to a pagan temple in Caesarea. Philo recorded this in Embassy to Gaius 38.299–305.

Pilate apparently killed a group of Galileans who had come to Jerusalem to sacrifice (Luke 13:1). We don't have any other information about this, but as Galileans, they would have been Antipas's subjects.

Emperor Tiberius's lieutenant had been Sejanus who was known for his cruelty and his hatred of Jews. It's possible once Sejanus was dead, Pilate felt free to be more open to a friendship with Antipas. This theory, however, doesn't seem to mesh with Pilate's long-time good relationship with Caiaphas the high priest.

Another theory is when Pilate wanted money from the temple treasury to build a waterway to Jerusalem, the Jews rebelled, and Pilate scattered assassins around a crowd who killed the Jews with daggers. Some think the Jews were Galileans and Antipas was angry Pilate killed his people, but it's unknown if that's true.

Whatever the cause, Pilate's invitation to have a say in Jesus' fate seems to thaw old animosity. Antipas humiliates Jesus but returns Him to Pilate with the same decision: Jesus is innocent (Luke 23:14–15). This makes their own guilt even more egregious as they fulfill Psalm 2:1–2:

"Why do the nations rage
and the peoples plot in vain?
The kings of the earth set themselves,
and the rulers take counsel together,
against the LORD and against his Anointed…"