Take a look at these two path; Illustrator says they have the same width profile, but it’s obviously not correct.
If saved as Graphic styles — these do match.
Paths match.
Directions match (and simple reversing still looks different).
The left-hand path cannot be edited with the native Width tool.

I can edit these with Width Selector tool by Astute Graphics, but it makes the width changed in a different spot. Saving the style, clearing the appearance, reapplying the style — heals the bug. Deleting a leftmost segment... this keep the look the same (almost), but the bug gets healed — the native tool now works. It’d be very interesting to know, how it was created — but I don’t know, actually.