Extrusion along a curve: position of the profile I want to make a curved strip with smooth edges.

I have made Bezier Circle and tweaked a bit to obtain something between an ellipse and a rectangle (this is the section of the strip). Then I made the curve that describes the shape of the strip, and pressed Object Data button in Properties window. In Bezier Object slot I selected the previously mentioned "ellipse".

The problem is that the ellipse appears rotated. The greater axis of the ellipse and the other curve should be coplanar, but they are perpendicular.

I have rotated the original ellipse, but this doesn't seems to have any effect on the final surface.
Without an image of some sort it's hard to guess, but this is a common enough problem that I think I know what you're facing.

If I understand you correctly, all you need to do is rotate the profile curve in edit mode, (instead of object mode) to align the axes of the profile with the axes of the other object.

The documentation is a little obtuse, but it might be helpful.