I have 3 working simultaneously without issue and in very close agreement, many of those early posts are on the forum.
3456a have no batteries, the constants are maintained by the setting of its pots. However, once you tweak it after being untouched for decades, it will drift quite badly in the uV ranges, and need to be retweeked until it settles back down; I suggest if you did adjusted it, wipe the adjusts screw up and down multiple times on a range as these old hermetically sealed pots could still have a lot of deposits on them, only then start to calibrate it to match the 34401a. Note, the mechanical adjusts affect the linearity on other ranges so do a performance check on all the ranges each time you move the pots.
The main disease in working 3456a units is the PSU, particularly bad old electrolytics, if they do not provide stable rail power, the readings will drift. Check the power rails for AC leakage, as well as incorrect output DC. If your disagreement is in the uV chances are its the PSU.
I agree fully with alm. Its also documented on the HP yahoo users group.
FWIW I found that semi-bad electrolytics in high quality old T&M gear can be 'reformed' by simply leaving the old device on 24/7 for 1 week or more after a PSU cap reform technique on first power up. The sign this works is the drifting stops and the output stabilizes, not just on the 3456a, but many other T&M left in storage somewhere. Ultimately its better to replace the caps, but 90% of my lab devices were salvaged that way.
https://www.nmr.mgh.harvard.edu/~reese/electrolytics/#reform