I've come across these terms "reversals allowed" and "reversals not allowed" in a sample drawing while reading a GD&T course.
What exactly are these terms? What do they mean?
It approximately means it is allowed or forbidden to have scratches at the place of round. When you are making cylinder surface you can to do it many different ways: milling, welding, soldering. In one case must be strong requriment for smoothness of surface (for example in bearings need to minimize friction). In other cases it is allowed to have scratches (for example any part of static construction, which is not contacted with any other surface).
Radius of an arc or circle, with no flats or reversals. This strict version of radius definition is specified in demanding applications when the form of the radius must be controlled more strictly than "just falling within the dimensional tolerance zone". CR is for critical features whose performance truly requires near-perfect geometry.
Imagine you were driving a car along the path shown in a cross section view of the radius - you need to make a 90 degree bend, and you need to stay inside the cones that denote the tolerance band.
In the first image, you find yourself about to hit the cones, so steer sharply left, then almost hit the other cones, so steer right, left, right etc.
In the second image, you are only able to turn the steering wheel left. Exactly how far left you turn it will define whether you’re closer to the inside or outside cones.
“Reversal” is when, in the first image, you move the steering wheel from turning left, through straight ahead, to turning right - reversing the curvature of the path you’re taking.
As Andrew has mentioned, in the real world this means no scratches (and consequent stress risers) in the fillet, but I hope my answer serves to more clearly illustrate the meaning of “reversal”.
In ASME Y14.5:2018 5.16.2, the term reversal is used in the definition of CR:
The part surface shall be a fair curve without reversals.
The term is not defined. It probably means that the curvature sign (measured in 'some' plane) doesn't change as figure shows.
For your information, in ISO GPS ('Geometrical Product Specification') tolerancing, the specification of radius is discouraged in ISO 14405-2:2011 7.5 and use of profile tolerance is recommended. The use of profile tolerance doesn't limit the 'waviness' of the curve. For that in ISO you would probably use filtration parameters.
PS: sorry, but are you allowed to disclose part of the course?