Weird normals problem / model display Unity

I’m building a short interactive game level for my workplace, and since this is my first time using Unity i’m a bit turned around.

Today I finished one of the main models for our simulation and imported it into Unity, only to have some very strange geometry problems. here’s a picture of it in Maya:
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Normals are all hardened on the container body / pretty much all the geometry (except for some fiddly bits on the door) and all facing the right direction. Normals are also facing the right direction.

Now here it is in Unity:
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Normals are all messed up, the other side of the container is apparently all smoothed out normal-wise, and on the shown side they’re doing god-knows-what. Plus, the inside of the container is see-through.

What am I doing wrong?

In the mesh import settings you can specify how/if normals should be calculated or imported. The normals directions should be the same as in Maya though, are you sure it is facing the right direction? And also remember that 3D modeling programs often render polygons as double sided whereas Unity does not.