Two ships pass one another at 0.5c. Do they both see their own clock as the fast clock and their own meter stick as the long meter stick? Or is there a fast clock and a slow clock; a long meter stick and a short meter stick with both ships agreeing which is which?

Imagine two ships with atomic clocks, a large display on their surface display the timing of the clock and the ability to accelerate very fast to 0.5c. The two ships separate by one AU with a signal directly between them.

When the flash of the signal reaches the ships both zero their clocks and one of the ship accelerates instantly to 0.5c. When the two ships pass one another (akin to jousting) they can both see each others clock output.

Is this situation symmetric? Do they both think their clock is fast and the other ship is slow? Or instead do they see a fast clock and a slow clock and they agree which is which?