I have a power transformer with 120 and 220 volt primaries for ~650 volt from the secondary. If I wanted lower voltage from the secondary, is there any reason I couldn't run my 120 volt line voltage into the 220 volt primary of the trafo and achieve the desired result?
Yes, that will work. The output voltage will be 120/220 = 55% of what it would be when the input is connected as intended. The only downside is that the overal power capability will be lower. If you can live with the lower maximum power output, then this will work fine.
There is no reason- stay within the current limit specified and you'll be fine, and you can get about half power out of the transformer.
Of course if you have a resistive load, and halve the voltage, the power will be reduced to 25% (or 29% in your case with 220/120).