Old wiring into appliance has 60V neutral to hot and 120 either to ground?
Maybe hot miswired to neutral somewhere and your hot wire's connection is somewhere open and floating.

Panel      Fubar         Box

Hot --------.   -------- Hot                    &lt;--.
             \                                     |
Neutral ----  '--------- Neutral     &lt;--.          | 60V
                                        | 120V     |
Ground ----------------- Ground      &lt;--'       &lt;--'


The 60V is probably an induced "ghost voltage".


  Typically this voltage reading may be as high as 50% of the energized voltage in the same proximity.


If things change radically when you move the wires in in the box, it might be that you have a loose connection in a wirenut or similar, plus some goofy connections.

A low impedance meter (e.g. one with a "low-Z" setting) might help with the diagnosis. I think I'd be turning power off and remaking the connections in the box and at any places upstream toward the panel.