When entering a nether portal the player doesn't exit facing the same direction, but there is some preservation.
Example: I have 2 nether portals next to each other as shown on the screenshot below. When I enter the right one facing north, I enter the nether facing east as the portals are on different axes (the one generated in nether was turned 90 degrees). Wanting to face west I created the portal on the left of the screenshot and entering it south I exit facing east once again. When I turn, for example, 45 degrees left when entering the portal, I am indeed turned that when I exit, from facing east. When I enter the portals backwards, then I am facing west. So it does preserve the direction, but doesn't explain the completely different orientation from those 2 portals.
But it gets even weirder, as I made another portal, on the same axis as the one in the nether and entered it facing west and once again exitted it facing east. I have destroyed and rebuilt the nether portal just in case it was some auto-generation bug but it keeps happening.