A convex mirror placed upside down or upright does not invert the image of its subject. A convex mirror bows outward, and therefore its focal point is behind its surface. If the focal point is at or behind the surface of a reflective object, then images observed in that object appear consistent with the viewer's orientation.
Convex objects like spheres are curved outward and are opposite to concave objects, which curve inward. According to Wikipedia, concave and convex mirrors are curved or parabolic mirrors that distort subjects, often by making them smaller or bigger.
Convex mirrors and other reflective surfaces, such as those found on buses or metal doorknobs, make objects in the mirror look small or farther away. This also means that a greater area can be seen, which is why side-view mirrors are often somewhat convex. Because all of the light rays bounce away from the center of the mirror, there is no way for them to flip or invert; light rays have to cross each other in order to flip the image of a subject. The reflection in a brass doorknob or an ornament ball does not flip even as the object is turned upside down.
In the case of concave mirrors, like the inside of a spoon, a subject appears upside down until that subject gets close enough. Concave mirrors create focal points in front of their reflective surfaces, and if a subject goes between that focal point and the mirror, the image appears upright because the light rays no longer cross. Like in convex mirrors, upside-down concave mirrors continue to show upside-down subjects unless the subject passes the focal point.