Tip: Both GDM and SDDM have startup scripts that are executed when X is initiated. For GDM, these are in /etc/gdm/, while for SDDM this is done at /usr/share/sddm/scripts/Xsetup. This method requires root access and mucking around in system configuration files, but will take effect earlier in the startup process than using xprofile.
Is this when the screen is locked or when you're logged out? Those are two different things and I suspect it's the latter. That's probably sddm and I suspect it can be fixed by using Wayland with it. Should be some option in /etc/sddm.conf or so.
usually monitors can be freely rotated. if yours can't, the back usually has a square vesa mount on the back and you can just take out the four screws and reattach it the way you like.