Electromagnetism is symmetric with respect to parity. That symmetry is broken by the convention we choose to use for defining the magnetic field vector. Aliens on another planet could define magnetic fields to point in the opposite direction compared to our definition. They would then use a left-handed rule $\textbf{F}=-q\textbf{v}\times\textbf{B}$ rather than our right-handed $\textbf{F}=q\textbf{v}\times\textbf{B}$. If you get in radio contact with these aliens and try to get them to tell you whether their definitions are the same as ours or opposite, you can't tell without some external reference point that tells them which hand you consider right.
What exactly determines the direction that a given charge will experience a force? I.e. why does a negative particle experience a force in one direction and not the other?
why is it that positive and negative particles experience a force in opposite directions?
You can express the rules in ways that don't refer to the magnetic field or its arbitrarily defined flippable direction. For example, parallel current-carrying wires attract each other if the currents are in the same direction. Such rules are independent of which charges you define as positive and which way you define the magnetic field.
When expressed in these ways that avoid the arbitrary conventions, these rules follow from special relativity. The classic presentation at the freshman physics level is in the textbook by Purcell.