It is typically sampled at the start of every frame, once every display refresh, so about 60 times a second.
I looked at the manual for this display, and on page 19 it gives the power up/power down sequence.
The pin35 you correctly identified as the pin that will flip the display, is designated SHLR.
However it seems, as I postulated, that this pin is polled at Startup as part of the Initialisation process, it is not constantly monitored subsequently.
HannStar does not specify which controller it uses, and the datasheet isn't exactly clear. In the image 6.5, SHLR and UPDN are not part of the startup or shutdown sequence, but are grouped with
"data include R0~R7, G0~G7, B0~B7, HSD, VSD, DCLK, SHLR, UPDN, DE [...] MODE, RSTB, STBYB, DITH". This implies they are continuously sampled, not sampled only at power-up.
The controllers I have used, do sample the polarity/mirroring pins for every frame, including the ST7262E43 I already mentioned. Some, like
ST7789V2, do not have such input pins, but do have a command to select orientation and mirroring of the display. The displays like
ER-TFT070IPS-4 with
ST7277 controllers (that do have such an input pin) do not export that pin to the FFC. These are for 800×480 displays, like the one you have, by the way; I mostly use smaller ones, 320x240 and 480x320 ones.
Perhaps I should have worded that differently, say
"in my experience,", instead of
"typically", but if you look at display modules with datasheets freely available (ILI9341, ILI9488, ILI9881, ST7701, ST7789, NV3052, NT35510), they definitely allow changing the orientation at runtime via a dedicated command. Perhaps it is an error for me to make any inferences from these to all display modules.

Simply put, in my experience across the various controllers, the direction is sampled at every new frame, and can be switched without turning the display off. It is quite rare for
anything to be only sampled by the display controller at power-on, and ignored afterwards: none of the displays I have do that.
So no amount of physical stimulation of the ribbon cables after boot up will be able to flip the display, or indeed poking about with a screwdriver as you imply.
Are you sure? Did you verify this in practice, or is this just how you understand the datasheet? In my opinion, the datasheet isn't clear on this. In particular, R0~R7, G0~G7, B0~B7, HSD, VSD, DCLK and DE are the digital video signals, and they definitely aren't simply sampled at power-up; they are continuously sampled, as otherwise the display would be all in single color.
I am not saying it is impossible, I am only saying it would be different to all the display controllers I have. Granted, I don't have
that many, maybe
a two dozen, but I do have considered and compared the most commonly available openly documented ones, and read their datasheets and pinouts, and examined the open source libraries used to drive these from microcontrollers (which is basically what I do for a hobby, as I've explained above). I myself have plenty of time for my hobby, but only a small budget.