What if I say that the mirror doesn't flip left and right? You've heard it right the mirror doesn't do the flipping. As the above answers say the mirror shows what is right infringe of it. It's you(we humans) who think it is flipping.

**Let me get this in detail**

Before we begin tell me ,'What makes you think that the mirror flips your left and right?'
Or may be 'what image do you it flips?'
The second question is a follow up question of the first one.

To answer these questions you need a camera not just an ordinary camera an imaginary,invisible camera
{ i.e i(camera) i,as in x+iy not in iPhone *pun intended*}

Now take a picture of yourself (No, not a selfie)with that camera at a distance x and at height of your eye from ground. What do you sees?
you will see a picture of 'beautiful you' in that picture with your left and right **unswapped**

Now place the mirror facing you at a distance x/2 from you while you are standing still ( now you know why I said,no selfie) 

Now take a picture of the mirror with the cam at your eye (I know it's practically not possible but trust me  this 'i' can do anything) what do you see? You will see the same 'you' as beautiful as you were in the first pic but with your left and right **swapped**

Now you see that the image is a flipped left and right.
(I guess you can answer the second question now)
Now, your question is why didn't the camera flip the up and down, right?
I'm getting there.

It's because of you. It because you did something when you were taking the second pic. You must be thinking 'what did I do? I barely moved'
After you inserted the mirror when I asked you take the second pic how you bring the cam from x you?
You brought the cam to you with its lens still facing you and **turned the camera around by 180 ° towards the mirror about the vertical axis** (bringing it to you and turning need not be in the same order) that's why the image is looking flipped left-right

Had you turned the camera 180 degrees about the horizontal axis we would have seen a vertically inverted image.

I hope you understand what turning about horizontal axis means.
When you turn the camera by 180° about horizontal the camera will face the mirror and the camera will be upside down