Three for the price of two?

Take a solid sphere. Its surface (or "skin") is homeomorphic to the sphere. Now drill a hole through the solid sphere:

The surface (or "skin") of the resulting solid is homeomorphic to the torus.

This time, drill two holes in the solid sphere, like this:

Now the surface (or "skin") of the resulting solid is homeomorphic to the 2-fold torus.

Question

Drill two holes in the solid sphere which "meet", like this:

To what is the surface (or "skin") of the resulting solid homeomorphic?

Notes

1. The torus and 2-fold torus were easy to recognise as the earlier answers. Most people cannot just "spot" the answer in this more complicated question. So some topological technique is needed.
2. You may assume the surface is orientable and has no boundary components.

Answer