Let me ask a related question. I will often read advice to sharpen one side till a burr is formed, then switch to the other side and repeat the process. But, at least in the case where a blade is dull and a fair amount of steel needs to removed to initially form a burr, aren't you creating an uneven bevel by following this advice? That is, while you would be doing many passes on the first side to get a burr, once that's done and you switch sides, it won't take much to re-form the burr (or flip it over). So, isn't the result an uneven bevel?
On a dull knife, wouldn't it make more sense to do a moderate number of passes on one side, then switch to the other side (even though a burr hasn't formed yet) and do a similar number of passes, and then switch back, and so on till the burr is formed, thus keeping the amount of steel removed from each side about the same?
Andrew
I think that's why some experienced knife sharpeners examine the knife first and one of the things they check is whether the bevel is uneven. If the bevel is uneven, then they would work on the appropriate side first. If the bevel was already uneven in the first place, your method of switching sides after similar number of passes would only maintain the unevenness of the bevel.
If the knife was a "working" knife, I don't believe the biggest concern is the unevenness but rather two things: 1) keeping it sharp and 2) extending the tool life. Given that, if the bevel was uneven, the side to sharpen first would be the one with the shorter bevel. Then the goal would be not to remove enough material so that the bevel would be even but, rather, to fully apex the edge.
Because the whole length of the bevel won't be shaved down at the same time to the point when that side forms 1/2 of an apexed edge and because it is quite difficult to stop the sharpening stroke at exactly the point when that side forms 1/2 of the apex and is just short of one molecule away from forming a burr, it is humanly impossible to not create a burr and have the that side form 1/2 of an apexed edge. The goal then is to remove enough material for that side to be 1/2 of the apexed edge and one evidence of this is that a burr is formed along the edge. Some people recommend to make a large burr to ensure that the bevel is indeed formed to be 1/2 of an apexed edge.
People use stropping as a final step to remove the burrs to "perfect" the edge. This tends to indicate that burrs do occur even if that isn't the main goal that people shoot for.
To answer your question: on a dull knife, if the bevels on either side were even, you would just work either one of the sides until you've removed the minimum material and have 1/2 of the apex. Then you'd work the other side. When you're done, the edge is sharp, the bevels remain even, and only the minimum material have been removed thus prolonging the tool life. If the bevels were uneven, you'd work on the short side first and do as above. The bevels may still not be even after the knife has been sharpened, but continued working the short side first on subsequent sharpenings will address the issue.