Who threw the suitcase to the other?

In Tenet we can clearly see that:

From the Protagonist perspective

I take the inverted suitcase
I throw it to inverted sator.
The inverted Sator catches the suitcase

From the perspective of inverted sator and inverted suitcase, the protagonist is the one who is inverted.

From Inverted Sator Perspective

I take the suitcase
I throw it to the inverted protagonist
The inverted protagonist catches it.

From Inverted suitcase perspective:

I am in the hands of Sator
He throw me to inverted protagonist .
The inverted protagonist catches me.

The question is: normally, the protagonist is the one who is supposed to throw the suitcase to Sator and not the other way around.

But since this movie presupposes that there is no "preferred" time direction, it follows that from some perspective: Sator is the one who threw and let go of the final piece of the algorithm.

What did I miss?

Thought: I think the clue is in the inverted suitcase, what would happen if the suitcase were not inverted. Maybe if a non-inverted person throws an inverted object to an inverted person, it counts as "giving", while throwing a non-inverted object to an inverted person counts as "taking" or giving from the perspective of the inverted person. My head is spinning already.

Best Answer

The suitcase was never inverted.

In forward movement, Protagonist (I'll call him P) throws it to ~Sator (I'm using ~ to denote an inverted person) and he catches it.

From ~Sator's POV, you will see that during the early part of his and ~P's chase, the orange box flies into the car ~Sator is in, so until the moment that exchange happened, Orange Box is in ~Sator's hands. So during the exchange, in his POV, he throws it to P.