Does it? It seems that a “hyperbolic” shape would be curved differently depending upon where you’re at . . . but maybe I’m wrong.
As you can see, a triangle drawn through the Universe has a different shape depending upon whether the Universe is a hypersphere, flat, or hyperbolic.
In the hyperbolic Universe, the triangle seems (at least to me) to change its shape if we turn it. This is what I mean by the asymmetry.