Hmmm, I never thought "Bent to one side". I've noticed that while I'm riding and I look straight down onto the frame, it looks like I'm leaning to one side. I tried to see if it was because I was on the right side of the crown of a road but even on level ground there is a slight lean. It tracks straight without hands. Could it be a fork issue?
It certainly could be, and probably is, a bent fork set.
I posted the front end picture of a bicycle that had bent forks, bent both backward and to the side. That bicycle, a Mercier, rode dead straight, hands off of the bars. It was not until I had built the bike, and then set it up, for picture taking, that I noticed the bent fork issue.
My guess is this old bike had both bent forks and a bent frame set, one canceling the negative ride qualities of the other out. Werid and that old Mercier now hangs in The Old Shed. Of course, now that I am becoming a fork fixing fella, I might have a go at those old French forks, and some Bianchi forks, and...