I want to know this out of curiosity
Forget "up" and "down". The laws of physics doesn't care about orientation.
Why does it "seem" that mirrors flip L and R? Your body has a mirror plane along the Z axis. When you rotate about your Z axis you almost look the same as if you're seeing yourself in the mirror. The differences are seen with your hands who have changed handeness (the actual technical term).
Note, if you rotated about your Y axis and did a hand stand you wouldn't be concerned about your feet being where your head was because your head and feet don't look alike.
So mirrors can't "flip it" side ways either, it just seems that way. Nor is it precise that the mirror "inverts" left and right. Instead it preforms an symmetry operation called, mirror. When you turn around 180 degrees, you're preforming another symmetry operation called... rotation (very clever naming. It's also known as a two-fold rotation). There is nothing inherently more correct about a rotation vs. mirror symmetry.