There are effectively three types. Linear polarisation is when light waves oscillate over one axis only. Picture an X-Y graph. Linear polarised light coming towards you will oscillate over 1 axis only, say the x-axis. Circular polarised lights oscillates over 2 axes, say the X and the Y axis, drawing a circle as it oscillates. Unpolarised light oscillates all over the X-Y graph.
Those are some interesting observations. I personally cannot use any IPS screens without developing migraines, but can happily use a few specific TN screens. So I am intrigued to hear they have different linear polarisation angles. Though, if polarisation angle were the only cause of my problems I might expect to get headaches whenever I tilt my head whilst using a TN.
Your observations that some screens do not go entirely black when rotating a polarising film: I think this implies they have some level of circular polarisation already. From what I understand, this may be a natural property of some plastic sheets - I skimmed a paper outlining how a specific brand of overhead projector transparency was found to have some quarter/half wave plate activity. Or perhaps it was a design choice by using wave plate film.
As I explained in the OP, OLED screens use polarisation to improve contrast against other light sources - ie to stop the screen looking washed out in sunlight. Turning up the brightness of an OLED screen to counter that is not ideal due to how rapidly OLEDs degrade. Hence the polariser solution.
This link explains the science of it. Reading it, it seems that OLEDs use circular polarising filters already. I will have to look out for a linear polarisation filter as first step to explore the polaristion of the devices I can and cannot use.