Consider the surface formed by taking a filled in square and identifying diagonal vertices. For example, $(0,0)\sim(1,1)$ and $(0,1)\sim(1,0).$
Image courtesy of @robjohn:
Is there a name for this particular surface? What can be said about the curvature?
Parametrization:
$$\scriptsize\left\{(x+y)\cos\left(\tfrac\pi4(x+y)\right)+(x-y)\cos\left(\tfrac\pi4(x-y)\right),(x+y)\cos\left(\tfrac\pi4(x+y)\right)-(x-y)\cos\left(\tfrac\pi4(x-y)\right),\tfrac\pi2\sin\left(\tfrac\pi2xy\right)\right\}$$
$x,y\in[-1,1]$.
There doesn't seem to be a name for this particular surface.
About the curvature, we can say that the surface, call it $S,$ has positive, negative, and zero curvature. This is due to a theorem which says:
Let $S\subseteq \mathbb{R}^ 3$ be a connected, regular, compact, orientable surface which is not homeomorphic to a sphere. Then $S$ has positive, negative and zero curvature.
To illustrate this, the Gaussian curvature is represented on the surface $S$ below.
Red=Negative, Green=Zero, and Blue=Positive.
Image courtesy of @robjohn.