You are seeing an image of an object in a curved mirror - the spoon.
You may find a soup spoon better than a dessert spoon as it is probably closer to being spherical?
First turn the spoon over so that it curves out towards you and look into it.
You will see an upright and diminished image of yourself in the spoon.
Here is the appropriate ray diagram.
What you will notice is that the image is distorted and this is because the spoon does not have a spherical (actually parabolic) shape and so produces lots of distortion (aberrations).
You will note that the image is formed behind the spoon and so is called a virtual image - one that cannot be projected onto a screen.
The problems with using the spoon are compounded by the fact that it has a very small "radius" of curvature and hence a small focal length.
Now turn the spoon over so that it is boring away from you and so acting like a convex lens.
You will see an inverted and diminished image of yourself which appears to be in the mirror.
It is not.
It is actually a real image which you a focussing on and it is in front of the mirror.
Just because there is no screen there does not mean that the image does not exist.
The image is there in space and your eye can collect the diverging rays from that image and focus them on the retina.
Now bring your face closer to the spoon and the image will get larger but then also fuzzier.
This is because the real image is now inside the near point of your eye, ie less that the least distance of distinct vision for the eye which is roughly 25 cm for the "average" eye.
Your spoon is also a useless shaving mirror.
If your face is within the focal point of the spoon you will get an upright and magnified image.
However the chances are that you will never be able to focus on this image because you eye will have to be very,very close to the mirror and observing the image of your eye will be almost impossible particularly as is inside the near point of your eye.
However if you place a pencil close to the mirror you should see a magnified image of the pencil.
For my spoon the pencil had to be about 5 mm from the spoon.
Proper shaving/make-up mirrors have a radius of curvature which might be hundreds of millimetres and so the natural position for the face is within the focal point of the mirror and hence an upright and magnified image is produced.
There was a question which I answered about the magnification produced by a shaving mirror which explains how you can measure the radius of curvature and hence focal length of your spoon.
You may be surprised that you do not need a screen to see a real image. There is an explanation as an answer to a question here.