I am not aware of any consumer direct view sets with user adjustable convergence, but I wouldn't say that "what you see is what you get". There are usually static convergence controls on the board; also convergence adjusting rings on the yoke of the picture tube. DIY attempts at adjustment are probably a bad idea, but an experienced technician with appropriate pattern generators, service manuals and plenty of patience can dial it in.
Unfortunately a number of direct views have strange convergence adjustments. Color TVs used to have "pots" on the boards that enabled you to adjust convergence, but now many of these sets have taped on mini-magnets that are physically placed on the tube. I don't know of anyone that has the ability to "adjust" these. I imagine you could unstick the magnet and reapply it, but that sure is far from a "science".
Those stick-on magnets are mostly to correct for defects (flexing and such) in the shadow mask. For those televisons that don't have adjustment controls on the video board, the convergence is adjusted with magnetic rings and tabs around the yoke of the picture tube, as well as by the tilting and positioning of the yoke assembly itself.
The adjustment is very much a science. If it weren't, the people in the factory doing the initial calibration would have a very frustrating job, indeed. (Randomly applying magnets until the picture looks good?)
Not a DIY job, but a good factory trained technician should be able to adjust it.
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