How did the cars get inverted in Tenet?

After re-watching Tenet on the small screen there is one thing really bugging me. In the car chase sequence, there appear to be two inverted cars: the silver Saab driven by the inverted protagonist, and the black Audi that Kat is trapped in.

The question is, why are these cars inverted? Who inverted them, when, and why?

One could argue that these cars are not really inverted at all and are merely being driven by inverted drivers. But they're driving backwards, so if the cars aren't inverted then they must be in reverse gear, which wouldn't make for such a great high-speed car chase. Plus the Audi un-breaks the wing mirror of another car, and the Saab un-explodes and then un-flips, so it really seems like they're inverted.

In the case of the silver Saab, the protagonist finds it outside of the turnstile facility, so it seems to make sense that it was inverted there and then parked outside. But it doesn't seem to make much sense for anyone to do this. The inversion must have been performed after the events that we see on screen. Since the turnstile has been captured by Ives and his team, that means they must have inverted it. This is probably the most reasonable explanation, but in that case I don't understand how Sator gets the algorithm, because the algorithm ends up in that car after the car chase (in forward time), so if that car belonged to the good guys all along then there's no chance for Sator to get hold of it.

The Audi is a deeper mystery. In its inverted timeline, it starts off parked in the middle of the motorway during a gunfight. Sator's people put non-inverted Kat and the protagonist into it. Then (in reverse time) the protagonist pushes the brake pedal, which for some reason causes the car to start accelerating. Later, inverted Sator and his inverted henchman climb into it and drive it to the turnstile facility, where it ends up. (We see it there at the start of the sequence.) But how did the Audi end up at the gunfight in the first place, in its own timeline? Someone must have inverted it and reverse-driven it there - but who did that, and why?

My guess is that these questions don't really have answers. My feeling is that the film probably just isn't consistent on this point, and we're supposed to accept that when an inverted person drives a car it makes the car inverted, even though that isn't the case with bullets etc. But it's also possible that an explanation is hinted at in the film, and someone might have been able to spot it, so it seems worth asking the question.