If the polarity of the incoming power is reversed you will still have
120 volts between black and white and the furnace will do exactly as the
OP described. I have seen it happen before-- more than once. Make sure
the black is hot and the white is neutral. Around here, about 99+% of
furnaces are connected with a pigtail and receptacle, so the polarity
can be accidentally switched easily if anything is done anywhere in the
circuit. Most furnaces will run ok during a/c use that way-- it will not
show up until the heat is fired up. Larry