Why is it not possible to make a circle with NurbsCurve and NurbsPath?

I can take the points of a NurbsCircle and mess them up and then put them back in their original locations and get a perfect circle again. I cannot arrange the points on a NurbsCurve or NurbsPath the same way to get a circle.

Also, what is the difference between a NurbsCurve and a NurbsPath?

Why is the Order U of a NurbsCurve limited to 4 while a NurbsPath can go up to 5?

As to the difference in question, this BSE quotation may help:

I know that Beziers, Nurbs curves and Paths have different controls:

Bezier handles have control over tangents, and the curve always touches the control point.

Nurbs curves don’t touch the control points, they just bend towards them. The curve doesn’t reach the first or last control points, but ends around the second/second-last point.

Paths are the same as Nurbs curves, except the curve continues to reach the first and last control points.

For more visit https://blender.stackexchange.com/questions/58709/what-are-the-practical-differences-between-beziers-nurbs-and-paths

In Edit Mode NURBS Order U is 2 to 6. I guess in Object mode better accuracy is not needed. And as Paths are used for animation this may be necessary.

It turns out “NurbsCircle” and “NurbsCurve” and “NurbsPath” are all the same type of object.

Each point on a “Nurbs _____” has a W attribute only accessible from the N Side panel. The 4 corner points of a “NurbsCircle” have their W set to 0.345 and the other points set to 1. This produces a circle.

NurbsCurve and NurbsPath objects are identical except NurbsPath objects have the Active Spline > Endpoint checkbox activated which does what is described above about whether or not the curve touches the first and last control points.