That video is not actual gameplay. I don't believe any of the ships are capable of turning that fast. If they are, it's via a lot of upgrades that your starting ship doesn't have.
The closest I've been able to get is roughly like this:
- Full forward throttle (maybe even boost/afterburner)
- Turn off flight assist
- Reduce throttle into the blue range (about 50%) or low end of the blue range (30-40%)
- Full pitch until you're turned around 180 degrees. More realistically: full pitch and a bit of yaw and/or roll until you're pointed at the enemy.
- Kill throttle.
- Reverse throttle.
- Re-enable flight assist.
Alternately, instead of reducing your forward throttle into the blue range, you can hit the up or down thrusters to keep your overall throttle in the blue range. If you simply leave your throttle off, you'll have a much slower pitch speed.
If you kill your throttle instead of dropping it, you'll stay traveling straight (more like the video), but your turn speed will be very slow.
Instead of re-enabling flight assist, you can reverse your pitch in order to level back. With flight assist off, the flight stick directly controls maneuvering thrusters and you have to provide the correct amount of counteracting thrust to stop. I find that disabling flight assist briefly is very helpful, but leaving it off makes it too hard to line up sights on things and tends to leave me tumbling in very hard to control ways.
If you try to yaw (turn sideways horizontally) instead of pitch (pull up/back or dive forward/down) you'll find that the turn speed is significantly slower. You could try doing both a pitch and a yaw maneuver at once, but I think you'd need to also have a bit of roll in there to get that to work out right.
I wouldn't try this up against the side of a space station (or asteroid, etc) or in close combat range. Where I have found this useful is to boost out of weapons range, flip backwards and then fly backwards at a speed the enemy can catch up while you do a bit of random pitch, roll and yaw maneuvers. Especially effective if you have gimbaled weapons, since it allows you to sustain fire at the incoming enemy while dodging their incoming fire. With fixed weapons you'll have to alternate between dodging and firing.