Meaning, about an axis perpendicular to the strip (or at least it's like that at one point--see below) and running through the center of the "hole" in the middle?

You can do this, sure. But because of the shape of the strip, it won't be perpendicular to the rotation axis everywhere.

Try this experiment: make a Mobius strip and pick a small area of it that you will define to be in the plane of rotation. Draw a mark on each side of the strip pointing "outward", i.e., away from the rotation axis.

Now take a small arrow and start it on one side of the strip pointing in the same direction as the little mark. Move the arrow around the strip. You will find that it will reach a point on the other side of the strip, where the other mark you drew is--and when it's there, the small arrow will be pointing in the opposite directon to the other mark. In other words, the small arrow is now pointing at the "inside" edge of the strip, not the "outside". (Note that on an orientable surface, you would not be able to move continuously along the surface from one mark to the other at all without crossing an edge.)

In other words, while yes, there is a well-defined "direction of centrifugal force" arrow everywhere in the space the strip is embedded in, it cannot always point in the same direction relative to the strip. In some places it points "horizontally" relative to the strip, towards one edge (but it's impossible to continuously pick "the same edge" everywhere on the strip relative to this). In other places it points "vertically", out of the strip. So the "direction of centrifugal force" does not give you a well defined orientation everywhere relative to the strip. You can't use it to say, for example, "this is the outer edge" and have that work continuously everywhere.