Why is it when I look into a spoon, my reflection is upside down?

Basically, because the spoon is curved inward. Rays from the top of your face will be reflected going downwards, rays from the bottom of your face will be reflected going upwards, creating an inverted image.

Reversal of image is the property of reflection of a concave mirror. (Inwardly curved is concave and outwardly curved is convex).

The spoon's surface is concave (curved inwards). Concave mirrors give reflections upside down, and strangely the image in the mirror appears to be in front of the mirror rather than behind it. This can be explained using complicated diagrams showing rays reflecting and converging, but at the end of the day the only simple answer is 'that's just the way it is'.

A difficult question to answer without a diagram. Basically the light rays that strike the top of the curve are reflected downwards and return as the bottom of the image. Similarly the light rays from your face that hit the bottom of the spoon are reflected upwards and return as the top of the image. Have a look in a physics school book and the diagrams should make it clear.

Because the concave shape of the spoon acts like the lens of your eye or a camera, i.e. it crosses light rays at a certain point so that the reflected image is upside down.

If you really want to mix yourself up, it would be right side up if your brain wasn't already flipping the image. The lens of your eye inverts images as well. When the image strikes your retina, it is upside down and your brain goes through the trouble of turning it right side up.

Lenses and mirrors work in opposite ways, though. Mirrors bounce light, and lenses bend them. The concave spoon is a mirror and bounces the light in. After the point where the beams cross, they are opposite. There is an exhibit in the Corning Glass Museum where you can walk up to a giant mirror, passing through the focus point. It'll give you a headache in a hurry. At the starting point, you see yourself upside down, and by the time you get up to the mirror you're right-side up again.

Your reflection on a spoon is upside down since light travels straight until it hits a curved surface reflecting a different angle

https://www.physlink.com/education/askexperts/ae43.cfm