Yup, thanks for the update.
Most panels are numbered; Top left #1, top right #2. Down the left side will be 1, 3, 5, and so on. Down the right side will be 2, 4, 6 and so on. On a single phase panel 1 and 2 are on phase A, 3 and 4 on phase B, 5 and 6 on phase A. The rows alternate between A and B phases.
You have said #3 and #5 were the two breakers feeding the same circuit but based on what I described above that would mean A phase and B phase were feeding the same circuit, which, would be a dead short.
Except, of course, on a Pushmatic panel, which has phase A on the left side & phase B on the right side.
Oddest looking double-pole breakers that you'll ever see.
It kind of sounds like it may be double fed.