No, you don't see your real face in the mirror.
The image you see is a reflection, which is a reversed image of your actual appearance. This is because light bounces off your face and then off the mirror's surface, creating a mirrored image.
Think of it like this: if you hold up your right hand, your reflection will show your left hand. This reversal applies to your entire face, including your features, like your eyes, nose, and mouth.
While you don't see your exact, "real" face, the reflection provides a reasonably accurate representation of your appearance. It's how you see yourself in everyday life and how others perceive you.
If you're curious about how your face actually looks, you can try taking a picture of yourself with the front-facing camera on your phone. This will show you a non-reversed image of your face.