Why does a camera distort your face?
I don't want to engage in the branch of Physics that deals with light, optics, etc., but the distortion you get comes from the design of a fixed lens. It is a complicated phenomenon, and to fully explain it, you would need to read a book on lens design and optics.
The lens is too close to the subject. Move them farther apart and see what happens. For any lens, there will be one specific distance at which image distortion will be minimized. This distance is different from one lens design to another.
You are seeing perspective distortion which happens when the camera is too close to the subject. It's just like taking a picture of a dog's nose when the camera is really close. To minimize perspective distortion, the camera needs to be at least 5, or more, feet from the subject. It's all about distance ratios. Taking selfies at arms length is sure to make you look clownish.
Quality photo portraits are made when the lens is not close to the face. Too close = distortion.
There are two types of distortion that I'd refer to as positive and negative distortion. This isn't an established term, just one that I made up to answer your question. Negative distortion, to me, describes the way a lens can distort the human face to make it look less attractive. Normally one would call this wide-angle distortion of the perspective. Wide-angle lenses distort the perspective making things closer to the lens look much larger than things at further distance. The distance between near & far elements doesn't have to be great. In fact use a wide lens close to the human face and it'll make the forehead, nose and chin look much larger than the rest of head. This is why so many people look bad in photos taken with smartphones. All smartphones have wide lenses that are equivalent to about 24mm~28mm in 35mm format. A "normal" lens is commonly a 50mm lens on a 35mm format (full frame) camera. This focal length produces a perspective very close to what the human eye sees. To be more exact, 43.9mm is closer to the perspective of the human eye. Once you go beyond 50mm, the perspective begins to compress. Near/far objects appear to be closer to each other. This has the effect of making people look more attractive. For this reason, lenses in the 85mm~200mm range are called portrait lenses. You can see the effect various focal lengths have on the human face here: http://www.stepheneastwood.com/tutorials/lensdistortion/tilepage.htm Now, as to "why" this happens, well, it obviously has to do with the way the optics in wide-angle lenses (a.k.a. short focal length) distort the light in order to fit a wider field of view into the area of the frame. This type of distortion is commonly referred to as barrel distortion. You can get more detailed information here: https://en.wikipedia.org/wiki/Perspective_distortion_(photography)
It isn't really distortion. Any perspective effect is due to the distance of the camera to the face. The image would look the same if you looked at a face with your eye at the same distance. Move your eye very close to someone's face and you'll see the same perspective effect. Also it has nothing to do with the focal length. The perspective effect doesn't change with focal length but with distance! Test: Take a photo with a wide angle lens. Then, from the same distance, another with a narrow angle lens. Or zoom in a zoom lens! The 'distortion' will be the same. Proof: crop the wide angle photo to have the same bits of photo as the zoomed in one, then enlarge it to be the same size. Both will look the same! Of course with a wider angle lens you'll have to move the camera closer to the face to fill the frame. So, for a lower 'distortion' photo you'd want the camera about 5 feet away. If you used a wide angle lens the face would be very small. So you have to use a narrow angle lens equivalent to 85mm. The reason one lens of your camera looks more distorted is that you move the camera closer to your face to fill the frame.
Frank is right. Depending on the aspect ratio of your camera, you need what would be the equivalent of 100mm lens, or longer, to get a good portrait. There used to be a principal in our local HS whose picture, when it was in the paper, looked like an old VW hubcap, The picture was taken on 35mm film with a 50mm lens. You see distortion a lot in the shows that have to do with renting apartments or flipping houses. often, the rooms look long like a bowling alley, but that is because of the wrong lens.
The back camera on Iphone flips the image.