Why does my FaceTime camera do this?

No. You're not going crazy. Mine does it as well. I have no idea why it does it so I'm probably of no help whatsoever other than to help keep your sanity in check... Sorry. On the upside though I have resorted to PS Express to correct the flip issue. Unless you have a hundred photos, it's as easy as swiping from left to right to get your picture back the way it was taken.
Update: The picture is initially taken AS a mirror image. Hold up anything with readable text and look at it on the display. It's backwards. When you take the picture its flipped and the text becomes readable again...

It renders properly later as I've discovered. You can test this by holding up a piece of paper with the word TEST. It will show in the camera TSET. After the shot and in Photoshop it'll say TEST. It's a natural looking error. Like holding your right eye shut would look like on the right side but it's supposed to be the left.

If you were using FaceTime and your face in the screen were not mirrored, it would be rather unnerving, as the face would not look like yours. We all think that we look like what we see in a mirror. 🙂