But if the surface has two sides, can it not? Consider the classic "gravity well" demonstration of a heavy ball placed on an elastic membrane. It forms a concave surface. Now turn your head upside down and look at the membrane from underneath. You'll see a convex surface. Same membrane, different curvatures.

You're thinking of this as someone who can look at the surface from the outside, in which case it makes sense to consider convex and concave curvatures to be different. But if you're restricted to moving within the surface, there's no way to tell the difference.

Within the surface, you can only measure the distances traveled along certain paths. For these distances, it doesn't matter whether you're walking across a hill or a valley of the opposite shape. The metrics of these surfaces are exactly the same.

That doesn't mean that negative curvature is impossible, just that it isn't as simple as inverting convex and concave. A negatively curved two-dimensional surface curves simultaneously in opposite directions, like a saddle point. But there's no way to have two surfaces with inverted curvatures at corresponding points, since curvature is determined by the arrangement of points, which means that inverting curvature obliterates the whole concept of "corresponding points" by arranging them differently.