Can I start a pivot, rotate 60 degrees, and stop a pivot in the same turn, and then fire weapons with no penalties?
This is permitted. You must of course fork over the thrust to both start and stop the maneuver, and suffer various maneuver restrictions (such as the fact that you can’t accelerate or decelerate). However, the rules actually state that the rolling, launch/land, and weapons fire restrictions are lifted after you stop pivoting.