Converting from Z-up right handed to Y-up left handed coordinate system?

I need to transform both vertex and normals between these coordinate systems:

OpenSceneGraph: X right, Y into the page, Z up
Unity: X right, Y up, Z into the page

As far as I can tell, changing between right and left handed coordinate systems can simply be acheived by swapping two axis. For example swapping z with y should be enough.

So my questions are:

1. What about the normals? Would swapping their axis be enough?
2. If I need to rotate the model after converting the normals to the correct coordinate system, do I simply apply that rotation matrix to the normal vector, as I do to the vertex point?