I think "reflect top to bottom" is intended to mean "swap top and button". A mirror reflects left, right, top and bottom perfectly.
It's front and back that it swaps.
Someone saying that a mirror swaps left and right is comparing it to a photograph, and only because we, as bipedal creatures, really prefer to orient images of other humans with heads up.
Someone saying that a mirror is swapped left and right is because they rotated themselves 180 degrees about the vertical axis to face the vertically aligned mirror. If they used a horizontal axis instead, they would have swapped top and bottom. And if the mirror is horizontally mounted on the floor, anything goes. You'd probably say it swaps up and down, which is front and back from the mirror's point of view.
Someone saying that a mirror swaps left and right is comparing it to a photograph, and only because we, as bipedal creatures, really prefer to orient images of other humans with heads up.