So yes, the hydro will "invalidate" the visual. Further, the hydro inspector should remove all stickers so to be able to do a proper inspection.
The scuba visual is a scuba industry standard which as part of this standard anytime a cylinder is completely empty of any pressure it should be re-inspected. The re-qualification process is required DOT is comprised of a hydrostatic test as well as visual inspection.
Now here is the gotcha - the visual inspection done by the hydro test is a valid visual inspection. Further the hydro stamp is proof of the visual inspection. However, hydro stations are typically not air fill stations. So it must leave the hydro shop and travel back to the dive shop for a fill. At that point the dive shop will perform another visual to bring up to their industry standard.
On the flip side, if the hydro station can do an air fill then you are set. However, getting a refill at a dive shop could be problematic because they are going to be looking for VIP sticker rather than a hydro stamp. And many do not believe that at hydro stamp is proof of visual inspection.
As for matching a hydro stamp with the visual. I never worry about that. Of course I inspect my our cylinders. Are you going diving soon?? If not just wait until you are then get the hyrdo.
BTW it is often cheaper to get the hydro yourself then take the cylinder to the dive shop for visual as some shops rightly will mark up the hydro cost for their time to take the cylinders to the hydro shop.
